A Group Theoretical Identification of Integrable Cases of the Li{e}nard Type Equation $ddot{x}+f(x)dot{x}+g(x) = 0$ : Part I: Equations having Non-maximal Number of Lie point Symmetries

A Group Theoretical Iden tification of Inte grable Cases of the Li ´ enard Typ e Equation ¨ x + f ( x ) ˙ x + g ( x ) = 0 : P art I: Equations ha ving Non-maximal Num b er of Lie p oin t Symmetries S. N. P andey ∗ Dep artment of Physic s, M otilal N ehru National Institute of T e chnolo gy, Al lahab ad - 211 004, India P . S . Bindu, M. Sen thilv elan and M. Lakshmanan † Centr e for Nonline ar Dynamics, Scho ol of Physic s, Bhar athidasan University, Tiruchir ap al li -620 024, India (Dated: Ma y 28, 2018) Abstract W e carry out a detailed Lie p oint symmetry g roup classification of the Li ´ enard typ e equation, ¨ x + f ( x ) ˙ x + g ( x ) = 0, w here f ( x ) and g ( x ) are arb itrary smo oth fu nctions of x . W e divide our analysis i nt o t wo parts. In the presen t fi rst part we isolate equations that adm it lesser parameter Lie p oint symm etries, namely , one, t wo and three parameter symm etries, and in the second p art w e iden tify equations that admit maximal (eigh t) parameter Lie-p oint symm etries. In the former case the in v ariant equatio ns form a family of in tegrable equations and in the latte r case they form a class of linearizable equations (under p oin t transformations). F urther, w e p ro v e the in teg rabilit y of all of the equations obta ined in the pr esen t pap er through equiv a lence transformations either by pro viding the general solution or b y co nstructing time indep endent Hamiltonians. S ev eral of these equations are b eing id en tified for th e fir st time fr om the group theoretical analysis. 1 I. INTR ODUCTION In t his set of tw o pa p ers we p erfo r m a L ie symmetry analysis for the Li ´ enard t yp e equation A ( x, ˙ x , ¨ x ) ≡ ¨ x + f ( x ) ˙ x + g ( x ) = 0 , (1) where o v er dot denotes differen tiation with resp ect to time and f ( x ) and g ( x ) are arbitrary smo oth f unctions of x . Notable equations from class (1) include a large num b er of ph ysically imp ortant no nlinear oscillators suc h as the anharmo nic oscillator, force-free Helm holtz oscil- lator, force-fr ee Duffing and D uffing-v a n der Pol oscillators, mo dified Emden-ty p e equation (MEE), and its hierarc h y , and generalized Duffing-v an der P ol oscillator equation hierarc h y . These equations arise natura lly in sev eral phy sical applications. The o utstanding represen- tativ e of the class of equations (1) is the mo dified Emden equ ation (also called P ainlev ´ e-Ince equation), ¨ x + αx ˙ x + β x 3 = 0, where α and β are arbitra ry parameters, whic h has receiv ed considerable atten tion from b oth mathematicians and ph ysicists for more than a cen tury (see for example R ef. 1 and references therein). During the past three decades, immense intere st has b een show n tow ards the searc h for symmetry g enerators of no nlinear ordinary differential equations (ODEs) and classifica- tion of low dimensional Lie algebras and linearization. Ev enthough L ie himself had sho wn that the second order OD E of the form, ¨ x + f ( t, x, ˙ x ) = 0, can admit a maxim um of eigh t symmetry generato r s, the recent imp etus came only when W ulfman and Wyb ourne 2 sho w ed that the maximal Lie group of p oin t transformations for the simple harmonic oscil- lator is eigh t and the a sso ciat ed gro up is S L (3 , R ). Subse quen tly , Cerv e ro a nd Villarro el 3 sho w ed tha t the damp ed linear harmonic oscillator also a dmits eight symmetry g enerators. Thereafter, sev eral studies we re made to isolate the equations whic h admit rich Lie p oin t symmetries b y exploring their symmetry a lg ebras and t heir applicatio ns in ph ysics and mathematics 4, 5,6,7,8,9,10,11,12,13,14 . F or a recen t surv ey of results one ma y refer Ref. 15 and references therein. Sp ecific equations of the form (1) ha v e also b een in v estigated from o t her p oints of view. F or ex ample, No ether symmetries for certain physic ally imp ortant systems w ere also studied in Refs. 3 ,1 6,17,18. Also con tact symmetries for the harmonic oscillator w ere also explicitly constructed by Cerv ero and Villarro el 3 . The nonlo cal symmetries for t he MEE hav e also b een studied in sev eral pap ers 19,20 . Recen tly m uc h in terest has also b een show n tow ards 2 exploring g eneralized symmetries, namely , λ - symmetries (also called c ∞ -symmetries) fo r certain nonlinear ODEs (see for example R ef. 21). The main g oal of the presen t set of pap ers is to presen t a detailed Lie p oint symme- try analysis of (1) and isolate in tegrable and linearizable cases explicitly . A t this p oint w e men tion here that the study of gro up classification is in teresting not only from a purely mathematical p oin t of view, but is also imp ortan t f or applications. Ph ysical mo dels are often constrained with apriori requiremen ts to symmetry pro p erties sp ecified b y ph ysical la ws, for example, f rom the Galilean or sp ecial relativit y principles. In this work we not only isolate the in v aria n t equations and symmetries but also presen t the integrals of motion and/or general solutions wherev er p ossible. The motiv ation for the presen t study comes from our r ecen t work in whic h w e ha v e considered a ra t her general second-order nonlinear ODE of the form ¨ x + ( k 1 x q + k 2 ) ˙ x + k 3 x 2 q +1 + k 4 x q +1 + λx = 0, where k i ’s, i = 1 , 2 , 3 , 4, λ and q are arbitrary parameters, and iden tified sev eral interes ting in tegrable cases and for certain equations w e hav e also derive d explicit solutions of b oth oscillatory and non- oscillatory t yp es 22 . In terestingly , we demonstrated that a system whic h is v ery close to the MEE, namely ¨ x + αx ˙ x + α 2 9 x 3 + λx = 0, p ossesses certain basic prop erties whic h is v ery uncommon to nonlinear oscillators 23 : It is a conserv ed Hamiltonian system ( o f nonstandard t yp e) a dmitting amplitude indep enden t harmonic oscillations. While the ab ov e men tioned equations are sp ecific examples of (1), the question arises as to whether there exist other in tegrable/linearizable second order O DEs b elonging t o this class. In this and the a ccom- pan ying pap er (referred to as I I), w e prese n t a systematic analysis to w ards this goal. In this w a y w e classify b oth integrable and linearizable equations which b elong to the Li´ enard t ype system (1 ). Ev en though the Lie’s algo r ithm is in principle a straightforw ard one, the group classi- fication for t he presen t problem is reduced to in tegration of a complicated o v erdetermine d system of partial differen tial equ ations for the infinitesimal symmetry functions for arbitrary forms of f ( x ) and g ( x ) in (1 ). While solving these determining equations w e hav e to choose all the symmetry functions (see Eqs. (12) and (1 3) b elo w) not equal to zero in order to obtain the maximal L ie p o in t symmetries. On the o t her hand considering the sp ecial case of one o r more of the symmetry f unctions to b e equal to zero also, w e obtain a sp ectrum of in tegrable equations. I n this w a y we are able to classify (i) systems with non-maximal symmetries (less than eight) (ii) systems with maximal symmetries (eight). No t e that for 3 second order ODEs the dimension of the L ie v ector space cannot b e fo ur, fiv e, six and sev en. In the presen t pap er w e fo cus our a tten tion only on the equations asso ciated with lesse r parameter Lie- p oin t symmetry groups, that is o ne, tw o and three parameter Lie- p oin t sym- metry groups. How e v er, within this classification, w e iden tify a wider class of imp ortan t in tegrable equations whic h are in v ariant under 2-par a meter p oin t symmetry gro up. Man y of them are b eing identified for the first time from the group theoretical analysis. In the second pap er w e iden tify all the equations admitting maximal num b er of symmetries whic h also turn out to b e linearizable. The forms of f and g whic h lead to only t w o symmetry generators are as follows: (i) f = k 2 + f 1 x q , g = k 2 2 ( q + 1 ) ( q + 2 ) 2 x + k 2 f 1 ( q + 2 ) x q +1 + g 1 x 2 q +1 , (2a) (ii) f = − f 1 + λ 2 log( x ) , g = g 1 x − ( λ 2 f 1 2 + λ 2 2 4 ) x log x + λ 2 2 4 x (log x ) 2 , (2b) (iii) f = f 1 x 2 , g = − A 2 4 x + Af 1 2 x + g 1 x 3 , (2c) (iv) f = λ 1 λ 2 + f 1 e − λ 2 x , g = − λ 1 λ 2 2 f 1 e − λ 2 x + g 1 e − 2 λ 2 x − λ 2 1 λ 3 2 . (2d) (Here f 1 , g 1 , k 2 , λ 1 , λ 2 , q , A are all constan ts). All the ab ov e equations are po in ted out to b e integrable through equiv alence transformations. W e also identify that the only system whic h admits a three pa r ameter symmetry group is the Pinney-Ermak o v equation where f = 0 and g = ω 2 x − ˜ g x 3 ( ω , ˜ g : constan ts). It is w ell kno wn that the infinitesimal generators of a giv en Lie group form a Lie algebra. The Lie alg ebras constituted b y Lie v ector fields a re widely used in the in tegration of dif- feren tial equations 5 , group classification of OD Es and PDES 24 , in geometric con trol theory and in the theory of systems with sup erp osition priniciples 25 and in differen t sc hemes for n umerical solution of differen tial equations 26 . A v ast amoun t of works is a v a ila ble in the literature on t he classification of realizations of finite dimensional Lie alg ebras on the real and complex planes. F or example, the realizations of all po ssible complex Lie algebras o f dimensions no greater than four w ere listed by Lie himself 27 . Recen tly Gonzalez-Lop ez et al ha v e provided t he Lie’s classification of realizations o f complex Lie algebras 28 and extended it t o the real case. A complete set of inequiv alen t realizations of real Lie algebras o f dimen- sion no greater than than four in the v ector fields on a space of an arbitr a ry (finite) nu m b er of v ariables w as constructed in Ref. 2 9. F or more details on the classification of Lie v ector fields one may refer the recen t w ork of Ref. 30 and reference s therein. On the ot her hand, in 4 our pap er we focuss our attention on constructing Lie v ector fields fo r t he class of equations resulting o ut of Eq. (1) alone and discuss their in tegrability . The plan of the pap er is a s follow s. In the following section, w e presen t the Lie’s algorithm for Eq. (1) and discuss the solv abilit y o f the determining equations. A careful a nalysis of our in v estigations show that one should consider tw o separate cases, namely (i) the symmetry function ( i ) b = 0 and (ii) b 6 = 0 , while solving the determining equations. Since the former case admits three symmetry functions, a ( t, x ) , c ( t, x ) , d ( t, x ), while classifying t he in tegrable equations, we consider the p ossibilities ( i ) d = 0 , a, c 6 = 0, and ( ii ) c = 0 , a, d 6 = 0 separately and bring out the equations that are in v ariant under b oth the p ossibilities in Sec. II I. W e also discuss sub-cases in b oth the cases ( i ) and ( ii ). F urther, w e prov e that the system (1) do es not admit a three parameter Lie p oin t symme try group when b oth f ( x ) , g ( x ) 6 = 0. In Sec. IV w e inv estigate the equiv alenc e transformations for Eq. (1) and show t ha t t hey lead to integrable forms. In Sec. V w e consider the sp ecial case in whic h either f ( x ) or g ( x ) is equal to zero and iden tify the asso ciated in tegrable equations in this class. The notable example in this class includes Pinney-Ermak o v equation. In Appendix A, w e presen t some details on the Ha miltonian structure of an integrable equation tha t arises in the case ( i ). In App endix B, w e p oint out briefly some notable equations that a r e included in the most general equation (corresp onding to (2a)). In App endix C, w e discuss the metho d o f solving the in tegrable equation iden tified as integrable in this pap er corr esp onding to (2 b). The L io uville in tegrability of the tw o other in tegrable equations iden tified in the catego r y c = 0 , a, d 6 = 0 are presen ted in App endices D and E. Finally , we presen t our conclusions in Sec. VI. I I. DETERMINI N G EQUA TIONS FOR THE INFI NITESIMAL SYMMETRIES W e consider the one dimensional nonlinear Li ´ enard type system o f the for m (1). Let the evolution equation b e inv ariant under the one parameter Lie group of infinitesimal transformations ˜ t = t + ǫξ ( t, x ) + O ( ǫ 2 ) , ˜ x = x + ǫη ( t, x ) + O ( ǫ 2 ) , ǫ ≪ 1 , (3) 5 where ξ and η represen ts the infinitesimal symmetries asso ciated with the v a riables t and x resp ectiv ely . The a sso ciated infinitesimal generator can b e written a s X = ξ ( t, x ) ∂ ∂ t + η ( t, x ) ∂ ∂ x . (4) Eq. (1 ) is in v aria nt under the action of (4) iff X (2) ( A ) | A =0 = 0 , (5) where X (2) = ξ ∂ ∂ t + η ∂ ∂ x + η (1) ∂ ∂ ˙ x + η (2) ∂ ∂ ¨ x (6) is the second prolo ng ation 5,6 in whic h η (1) = ˙ η − ˙ x ˙ ξ , η (2) = ¨ η − ˙ x ¨ ξ − 2 ¨ x ˙ ξ , (7) and dot denotes total differen tiation. By analysing Eq . ( 5) w e get the follow ing determining equations: ξ xx = 0 , (8) η xx − 2 ξ tx + 2 f ξ x = 0 , (9) 2 η tx − ξ tt + f ξ t + 3 g ξ x + η f x = 0 , (10) η tt − ( η x − 2 ξ t ) g + f η t + η g x = 0 , (11) where subscripts denote partial deriv ativ es. Solving Eqs. (8) a nd (9) w e obtain ξ = a ( t ) + b ( t ) x (12) and η = ˙ bx 2 − 2 b ℑ ( x ) + c ( t ) x + d ( t ) , (13) where ℑ x = F ( x ) = Z x 0 f ( x ′ ) dx ′ and ℑ xx = f ( x ) , (14) 6 and a ( t ), b ( t ), c ( t ) and d ( t ) a re a rbitrary functions of t . With t hese forms of ξ a nd η , Eqs. (10) and (11) can b e rewritten as ( ˙ bx 2 − 2 b ℑ + cx + d ) f x + ( ˙ a + ˙ bx ) f + 3 bg − 4 ˙ bF + 3 ¨ bx + 2 ˙ c − ¨ a = 0 , (15) and ( ˙ bx 2 − 2 b ℑ + cx + d ) g x − ( c − 2 ˙ a − 2 bF ) g − 2 ¨ b ℑ +( ¨ bx 2 − 2 ˙ b ℑ + ˙ cx + ˙ d ) f + ... b x 2 + ¨ cx + ¨ d = 0 , (16) resp ectiv ely . Solving Eqs. (15) and (16) for the giv en for ms o f f ( x ) and g ( x )w e can get the infinitesimal symmetries. The foremost and simplest solution for (15) and (16) for an y form of f and g is a = constan t , b, c, d = 0 . In ot her w ords, one immediately gets the time translation generator X = ∂ ∂ t irresp ectiv e o f the fo r m of f and g . Our mo t iv ation here is to find explicit forms o f f and g whic h admit mo r e nu m b er of symmetries. F or this purp ose w e solv e Eqs. (15) and (16) in the fo llo wing w a y . Rewriting Eq. (15), w e get g = 1 3 b [ − ( ˙ bx 2 − 2 b ℑ + cx + d ) f x − ( ˙ a + ˙ bx ) f + 4 ˙ bF − 3 ¨ bx − 2 ˙ c + ¨ a ] , b 6 = 0 . (17) Th us the existence of Lie p oin t sy mmetries of the general form (12) and (13) with b 6 = 0 in tro duces a n in terrelation b et w een the functions f and g . Ho w ev er, this relation has to b e compatible with the second determining equation (16 ). Th us using (17) into (16), one can obtain an equation for f , (whic h also in v olv es all the four symmetry functions) whic h fixes its form a s w ell as the ass o ciated symmetries sy stematically . This is carried out in the follo wing pap er II, where w e sho w explicitly tha t maximal (eigh t) num b er of Lie p oint symmetries exists only for the case f xx = 0 and for f xx 6 = 0 necessarily requires the symmetry function b = 0. Consequen tly , one has to consider the case b = 0 in Eqs. (15) - (16) separately b ecause of the condition b 6 = 0 in Eq. (17). Th us it is of in terest to consider t w o separate cases asso ciated with Eqs. (15 )-(16): Case (i) b = 0 : Since we assume one o f t he symmetry functions to b e zero the determining equations lead us to lesser Lie p oint symmetries alone (one, t w o and three symmetries). Case (ii) b 6 = 0 : In this case w e solve the full dete rmining equations whic h in turn lead us to the maximal (eight) Lie-p oin t symmetry gr oup (as w ell as other lesser p o in t symmetries while b 6 = 0) when f xx = 0. 7 In this pap er, w e analyze in detail only the Case (i) and presen t the results of the other case in t he subsequen t pap er I I. A. Alt ernate W ay One may also note here that one can pro ceed in an alternate w a y to analyze the deter- mining equations (15) and (16) for compatibilit y to determine the forms o f f ( x ) and g ( x ) and the asso ciated symmetries. F or example, Eq. (15) can b e rewritten as ¨ a = ( ˙ bx 2 − 2 b ℑ + cx + d ) f x + ( ˙ a + ˙ bx ) f + 3 bg − 4 ˙ bF + 3 ¨ bx + 2 ˙ c. (18) Differen tiating the a b o v e equation with resp ect to x and rearr a nging one can express ¨ b in terms of ˙ a, ˙ b, c and d . Differen tiating again the resultan t equation with respect to x and simplifying the latter w e find ˙ af xx = − [( ˙ bx 2 − 2 b ℑ + cx + d ) xx f x − 2 ( ˙ bx 2 − 2 b ℑ + cx + d ) x f xx ( ˙ bx 2 − 2 b ℑ + cx + d ) f xxx − ˙ bxf xx − 3 bg xx + 2 ˙ bf x . (19) Similarly from Eq. (1 6) one can obta in expres sions for ¨ c and ¨ d a nd finally arriv e at an expression of the fo rm 0 = − [( ˙ bx 2 − 2 b ℑ + cx + d ) g x − ( c − 2 ˙ a − 2 bF ) g − 2 ¨ b ℑ +( ¨ bx 2 − 2 ˙ b ℑ + ˙ cx + ˙ d ) f + ... b x 2 ] xx . (20) T o find a compatible solution of Eqs. (19) and (2 0) one may substitute for ˙ a from (1 9) in to (20) and analyze the resultan t equation to find the allow ed forms o f f and g and the asso ciated symmetries. How ev er, in practice w e find the metho d leads to very length y and lab orious calculations. On the o ther hand in the pro cedure w e adopt w e find that for the case f xx 6 = 0 , one necessarily requires b = 0, see Sec. V in the following pap er I I. Consequen tly the analysis giv en in Sec. I I I follows naturally . On the ot her hand, for the case f xx = 0, Eq. (19) leads to t he condition 0 = 2 bf f x − 3 bg xx . (21) whic h is consisten t with (20). F rom (21) one can immediately write f = f 1 + f 2 x and g = 1 3 f 1 f 2 x 2 + 1 9 f 2 2 x 3 + g 1 x + g 2 . T o fix the corresp onding symmetries one has to resubstitute the forms of f and g in the o riginal determining equations and solv e them consisten tly . This is carried out in P ap er I I. 8 I I I . LIE SYMMETRIES OF LI ´ ENARD T YPE SYSTEMS - LESSE R P ARAMETER SYMMETRIES: C ASE b = 0 W e now explore the nature of the ev olution equations whic h p ossess lesser parameter Lie p oin t symmetries. Considering Eqs. (15) and (16), w e now assume the f unction b = 0 to obtain the follo wing determining equations, ( cx + d ) f x + ˙ af + 2 ˙ c − ¨ a = 0 (22) and ( cx + d ) g x − ( c − 2 ˙ a ) g + ( ˙ cx + ˙ d ) f + ¨ cx + ¨ d = 0 , (23) resp ectiv ely . W e not e here t ha t the determining equation (22 ) for the function f do es not in v olv e the function g . As a consequence an explicit f o rm for f can b e determined b y direct in tegration. No w substituting this form of f in to equation (23) w e can deriv e the corresp onding fo r m of g . It is a w ell kno wn f a ct 5,6 that a second order ODE admits only 1 , 2 , 3 or 8 parameter Lie p oin t symmetries (whic h w e will see explicitly for Eq. (1) also in the presen t as w ell as in the follow up pap er I I). In Sec.2 w e noted that the most general equation whic h is in v ariant under the o ne parameter L ie p oint symmetry gro up is the general equation (1) itself with arbitrary form of f ( x ) and g ( x ) since there is no explicit app earance of t in t he equation and the asso ciated symmetry generator is ∂ ∂ t . But fo r b = 0, w e explicitly sho w in the follow ing that o nly t w o parameter symmetrie s exit when b oth f 6 = 0 and g 6 = 0 and obtain their sp ecific forms, while three parameter symmetries can also exist only when f = 0 , g 6 = 0. Finally , in the case f 6 = 0 , g = 0, the second order ODE (1) can b e rewritten as a first order equation whic h in turn can b e integrated b y quadratures straightforw ardly . So w e do not in v estigate this last category in this w ork. A. 2-pa ra meter Lie p oin t symmetries Rewriting Eq. (22), we ha v e f x + ˙ a c ( x + d c ) f = ¨ a − 2 ˙ c c ( x + d c ) . (24) 9 Since f should b e a function of x alo ne (vide Eq. (1)), we c ho ose ˙ a c = λ 1 , ¨ a − 2 ˙ c c = λ 2 , d c = λ 3 , (25) where λ 1 , λ 2 and λ 3 are constants. Then Eq. (24) b ecomes f x + λ 1 ( x + λ 3 ) f = λ 2 ( x + λ 3 ) . ( 2 6) In tegrating Eq. (26) o ne obtains f = λ 2 λ 1 + f 1 ( x + λ 3 ) − λ 1 , (27) where f 1 is a lso an arbitrary constant. Note that since f 1 , λ 1 , λ 2 and λ 3 no w o ccur in t he form for f ( x ) whic h determine the OD E (1), they a r e not symmetry para meters but rather they are the system p ar ameters . One can pro ceed further by constructing the asso ciated form of g b y solving Eq. (23) and classify the in v ariant equations. T o start with, let us first classify fo r con v enien ce the equations whic h are in v aria n t under 2-par a meter Lie-p oin t symmetry g roup. T o deduce these equations, we consider the t w o p ossibilities ( i ) d = 0 ; a, c 6 = 0, ( ii ) c = 0; a, d 6 = 0 and also the sub-case s in b oth of t hem. W e further note that due to the fact t he sys tem alw a ys admits translational symmetry , from Eqs. (1) and ( 1 2), it is clear that a cannot b e zero. So w e need not consider the case a = 0, c, d 6 = 0. On the other hand if b oth c a nd d are sim ultaneously zero, then g = 0 as ma y b e inferred from Eq. (23). Finally , at the end o f this section w e consider the cases where none of the functions a, b, c are zero and sho w that ev en here only 2 -parameter symmetries exist. 1. Case 1 d = 0 ; a , c 6 = 0 ( λ 1 6 = 0 , λ 2 6 = 0 ) The c hoice d ( t ) = 0 with a ( t ) , c ( t ) 6 = 0 leads us to sev eral intere sting new in tegrable equations, as w e see b elo w. Solving Eq. ( 25), with d ( t ) = 0 a nd so λ 3 = 0, one can obta in explicit for ms for the functions a and c as a = a 1 + λ 1 λ 2 ( λ 1 − 2 ) c 1 e ( λ 2 λ 1 − 2 ) t , c = c 1 e ( λ 2 λ 1 − 2 ) t , (28) where a 1 and c 1 are t w o arbitrary (symmetry) parameters, whic h lead to a t w o parameter Lie-p oint symmetry group. Substituting Eqs. (27) and (2 8) in to (23) with d = 0, we get g x + (2 λ 1 − 1 ) x g = − 2 λ 2 2 ( λ 1 − 1 ) λ 1 ( λ 1 − 2 ) 2 − λ 2 f 1 ( λ 1 − 2) x − λ 1 . (29) 10 In tegrating Eq. (29), we obtain g = λ 2 2 (1 − λ 1 ) λ 2 1 ( λ 1 − 2 ) 2 x + λ 2 f 1 (2 − λ 1 ) λ 1 x 1 − λ 1 + g 1 x 1 − 2 λ 1 , (30) where g 1 is a no ther integration constant. The ab ov e forms of f and g (vide Eqs. (27) and (30), resp ectiv ely) fix Eq. (1) to the form ¨ x +  λ 2 λ 1 + f 1 x − λ 1  ˙ x + λ 2 2 (1 − λ 1 ) λ 2 1 ( λ 1 − 2 ) 2 x + λ 2 f 1 (2 − λ 1 ) λ 1 x 1 − λ 1 + g 1 x 1 − 2 λ 1 = 0 . (31) F or the sake of neatness, w e rewrite λ 1 = − q , q 6 = 0, and λ 2 = − k 2 q , where k 2 is an a rbitrary parameter, in the a b o v e equation so that w e obtain ¨ x +  k 2 + f 1 x q  ˙ x + k 2 2 ( q + 1 ) ( q + 2 ) 2 x + k 2 f 1 ( q + 2 ) x q +1 + g 1 x 2 q +1 = 0 , (32) where k 2 , f 1 and q are nothing but system pa r a meters. Eq. (32) is the most general equation that is inv arian t under the tw o parameter Lie p oint symmetry group with infinitesimal symmetries ξ = a 1 − 1 k 2 ( q + 2 ) c 1 e k 2 q ( q + 2 ) t , η = c 1 e k 2 q ( q + 2 ) t x. (33) The corresp onding infinitesimal generators read X 1 = ∂ ∂ t , X 2 = e k 2 q ( q + 2 ) t  − ( q + 2 ) k 2 ∂ ∂ t + x ∂ ∂ x  . (34) The commutation relation b et w een the v ector fields X 1 and X 2 is give n b y [ X 1 , X 2 ] = k 2 q X 2 ( q + 2 ) . (35) 2. In tegrabilit y of E q. (32) for arbitrary v alue s of q Eq. (32) is the most general in tegrable equation whic h is in v ariant under the t w o pa- rameter symmetry group (33). No w we discuss the in tegrabilit y of (32) briefly here. By in tro ducing the transformatio n w = xe k 2 ( q +2) t , z = − ( q + 2 ) q k 2 e − qk 2 ( q +2) t (36) 11 where w and z are new dep enden t and indep enden t v a riables, resp ective ly , one can transform (32) to the form w ′′ + α w q w ′ + β w 2 q +1 = 0 , (37) where α = ( q + 2) 2 f 1 2 k 2 2 q 2 and β = ( q + 2)4 g 1 4 k 4 2 q 4 2 . Eq. (37) has b een analyzed from differen t p ersp ectiv es. F or example, Lemmer and Leac h 20 ha v e studied the hidden symmetries of Eq. (37). F eix et al. 31 ha v e show n that through a direct transformatio n to a third order equation the ab ov e Eq. (37 ) can b e in tegrated to obtain the general solution for the sp ecific choice of the parameter β , namely β = α ( q + 2) 2 . F or this choice of β , the general solution of (37) can b e written as x ( t ) =  (2 + 3 q + l 2 )( t + I 1 ) l l ( t + I 1 ) q +1 + (2 + 3 q + q 2 ) I 2  1 q , I 1 , I 2 : arbitrary constants . (38) F or the same parametric c hoice recen tly w e hav e sho wn that this equation can b e linearized to a fr ee particle equation t hrough a generalized linearizing transformation so that the solutio n of the nonlinear equation can b e constructed from the solution of the linearized equation 41 . Ho w ev er, our ve ry recen t studies sho w that Eq. (37) admits time indep endent Hamiltonian description for all v a lues of α and β . By in tro ducing appropr ia te canonical transformation to the Hamilton’s canonical equation o f motion o ne can integrate the r esultan t equations straigh tforw ardly a nd obta in the g eneral solution (for more details one may see Refs. 22, 34). F or con v enienc e, the Hamiltonian structure of the ab ov e equation is indicated in the App endix A. W e also p oint out briefly other notable equations included in (32) in App endix B. 3. In tegrable equations with d = 0 , a, c 6 = 0 ( λ 1 = 0 ( q = 0) , λ 2 6 = 0 ) Earlier while deriving t he form (2 7) f o r f ( x ) we assumed that λ 1 6 = 0. Now let us consider the case λ 1 = 0. F rom Eq. (25) we find that in this case a = a 1 , c = c 1 e − λ 2 t 2 , (39) where a 1 and c 1 are tw o arbitrar y symmetry parameters whic h ag ain lead us to a t w o parameter symmetry group. Solving Eq. (24), with the ab ov e forms of a and c , w e o bta in f ( x ) = − f 1 + λ 2 log( x ) , (40) 12 where f 1 is an in tegration constant. Substituting Eq. (4 0) in to Eq. (23) with d = 0, w e get g x − g x + λ 2 2 ( f 1 − λ 2 log x ) + λ 2 2 4 = 0 . (41) In tegrating Eq. (41), we obtain the following sp ecific form for g , g = g 1 x − ( λ 2 f 1 2 + λ 2 2 4 ) x log x + λ 2 2 4 x (log x ) 2 , (42) where g 1 is a n inte gration constan t. Using Eqs. (40) and (42) in Eq. (1), we ha v e the follo wing nonlinear ODE, ¨ x + ( − f 1 + λ 2 log( x )) ˙ x + g 1 x − ( λ 2 f 1 2 + λ 2 2 4 ) x log x + λ 2 2 4 x (log x ) 2 = 0 , (43) whic h is in v ariant under the following infinitesimal symmetries ξ = a 1 , η = c 1 e − λ 2 t 2 x. (44) The asso ciated symmetry generators t a k e the for m X 1 = ∂ ∂ t , X 2 = e λ 2 t 2 x ∂ ∂ x . (45) The in tegrabilit y of Eq. (43 ) can b e pro v ed straigh tforw ardly whic h w e indicate in App endix C. 4. In tegrable equations with d = 0 , a, c 6 = 0 ( λ 1 6 = 0 , λ 2 = 0) While deriving (27) w e assumed that λ 2 6 = 0. Now w e a na lyse the case λ 2 = 0 with λ 1 6 = 0. In this case w e find that the compatible solution exists for either λ 1 6 = 2 or λ 1 = 2 . In the first case b y repeating the previous analysis w e find that f = f 1 x − λ 1 and g = g 1 x (1 − 2 λ 1 ) , where f 1 and g 1 are tw o ar bitr ary parameters so that Eq. ( 1 ) b ecomes ¨ x + f 1 x − λ 1 ˙ x + g 1 x (1 − 2 λ 1 ) = 0 . (46) The asso ciated infinitesimal generators turn out to b e X 1 = ∂ ∂ t , X 2 = t ∂ ∂ t + x λ 1 ∂ ∂ x . (47) Eq. (46) exactly coincides with (37) by redefining λ 1 = − q , and so the inte grability of (46) can b e extracted fro m (37). 13 In the second case, namely , λ 1 = 2, w e obtain that the follow ing for m of equation for ( 1 ), ¨ x + f 1 x 2 ˙ x − A 2 4 x + Af 1 2 x + g 1 x 3 = 0 , (48) where A is an arbitrary parameter, whic h is in v a rian t under the t w o parameter infinitesimal symmetry generato r s, X 1 = ∂ ∂ t , X 2 = e − At  − 2 A ∂ ∂ t + x ∂ ∂ x  . (49) W e discuss the in tegrabilit y of Eq. (49) in App endix D. In this and previous sub-sections w e discussed the cases ( i ) λ 1 = 0 , λ 2 6 = 0 and ( ii ) λ 2 = 0 , λ 1 6 = 0 . Finally , f o r the third case, namely ( iii ) λ 1 = 0 , λ 2 = 0 , one gets a = constant = a 1 , c = constant = c 1 . The in v a rian t equation turns out to b e the linear damp ed harmonic oscillator equation ¨ x + f 1 ˙ x + g 1 x = 0, where f 1 and g 1 are arbitrary parameters. The asso ciated infinitesimal v ector fields are X 1 = ∂ ∂ t , X 2 = x ∂ ∂ x . It is kno wn that damp ed harmonic oscillator equation admits eigh t para meter symmetry gro up. Sinc e o ne of the symmetry f unctions is zero we obtained only a t w o parameter symmetry g roup. The full symmetry gr oup of the damp ed har mo nic o scillator will b e discussed in pap er I I. 5. Case 2 c = 0 , a , d 6 = 0: Integrable equat ion In the previous sub-section w e considered the case d = 0 , a, c 6 = 0. Now we fo cuss our atten tion on the case c = 0 , a, d 6 = 0 a nd fix the forms of f a nd g whic h ar e in v a rian t under the corresp onding symmetry transformations. Restricting to c = 0 and a, d 6 = 0 in (22), w e ha v e f x + ˙ a d f = ¨ a d . (50) As b efore, since f has to b e a function o f x alo ne, w e c ho ose ¨ a d = constant = λ 1 , ˙ a d = λ 2 = constan t , (51) so that Eq. (5 0) b ecomes f x + λ 2 f = λ 1 . (52) Solving (51 ) w e o btain a = a 1 + λ 2 2 λ 1 d 1 e λ 1 λ 2 t , d = d 1 e λ 1 λ 2 t , (53) 14 where a 1 and d 1 are t w o in tegration constan ts whic h are also the t w o symmetry parameters. In tegration o f Eq. (52) leads us to f = λ 1 λ 2 + f 1 e − λ 2 x , (54) where f 1 is an ar bit r a ry constan t. Substituting (54) into (23), with c = 0, w e get g x + 2 λ 2 g + λ 1 λ 2 f 1 e − λ 2 x + 2 λ 2 1 λ 2 2 = 0 . (55) In tegrating (5 5) w e obtain g = − λ 1 λ 2 2 f 1 e − λ 2 x + g 1 e − 2 λ 2 x − λ 2 1 λ 3 2 , (56) where g 1 is a n in tegration constan t. Eqs. (54) a nd (56) fix the equation (1) to the sp ecific form ¨ x +  λ 1 λ 2 + f 1 e − λ 2 x  ˙ x − λ 1 λ 2 2 f 1 e − λ 2 x + g 1 e − 2 λ 2 x − λ 2 1 λ 3 2 = 0 . (57) W e note that in the abov e λ 1 , λ 2 , g 1 , and f 1 are system parameters. Eq. (57) is in v ar ia n t under the follo wing tw o parameter Lie p oin t symmetries ξ = a ( t ) = a 1 + λ 2 2 λ 1 d 1 e λ 1 λ 2 t , η = d 1 e λ 1 λ 2 t , (58) where d 1 and a 1 are the symmetry parameters. The asso ciated infinitesimal generato r s are X 1 = ∂ ∂ t , X 2 = e λ 1 λ 2 t  λ 2 2 λ 1 ∂ ∂ t + ∂ ∂ x  . (59) The in tegrabilit y of the Eq. (57) is discussed in App endix E. B. Non-existence of 3-parameter symmetry group in the general case a, c, d 6 = 0 Finally , we consider the general case in whic h none of the functions a, c and d are zero. In this case, the function f tak es the form giv en in Eq. (27). The functions a, c a nd d can b e fixed b y solving the Eq. (25). Doing so w e find a = a 1 + λ 1 ( λ 1 − 2 ) λ 2 c 1 e λ 2 ( λ 1 − 2) t , c = c 1 e λ 2 ( λ 1 − 2) t , d = λ 3 c 1 e λ 2 ( λ 1 − 2) t , (60) where only a 1 and c 1 are the symmetry para meters. So in this case also only 2-parameter symmetries exist and no 3-para meter symmetry gro up is p ossible. 15 Substituting the forms f , a, c and d into Eq. (23) and simplifying the resultan t equation one obtains g x + (2 λ 1 − 1 ) ( x + λ 3 ) g = 2 λ 2 2 (1 − λ 1 ) λ 1 ( λ 1 − 2 ) 2 + λ 2 f 1 (2 − λ 1 )( x + λ 3 ) λ 1 , (61) where g 1 is a n in tegration constan t. One can directly integrate (61) to obtain g =  λ 2 λ 1  2 (1 − λ 1 ) (2 − λ 1 ) 2 ( x + λ 3 ) +  λ 2 λ 1  f 1 ( x + λ 3 ) (1 − λ 1 ) (2 − λ 1 ) + g 1 ( x + λ 3 ) (1 − 2 λ 1 ) . (62) Inserting the forms (27 ) and (62) in (1) we get ¨ x +  λ 2 λ 1 + f 1 ( x + λ 3 ) λ 1  ˙ x +  λ 2 λ 1  2 (1 − λ ) (2 − λ 1 ) 2 ( x + λ 3 ) +  λ 2 λ 1  f 1 ( x + λ 3 ) (1 − λ 1 ) (2 − λ 1 ) + g 1 ( x + λ 3 ) (1 − 2 λ 1 ) = 0 . (63) It is intere sting to note that the system p ossesses only tw o parameter Lie p oint symme- tries. The infinitesimal symmetries and generator s are ξ = a 1 + λ 1 ( λ 1 − 2 ) λ 2 c 1 e λ 2 ( λ 1 − 2) t , η = c 1 e λ 2 ( λ 1 − 2) t ( x + λ 3 ) , (64) and X 1 = ∂ ∂ t , X 2 = e λ 2 ( λ 1 − 2) t  λ 1 ( λ 1 − 2) λ 2 ∂ ∂ t + ( x + λ 3 ) ∂ ∂ x  (65) resp ectiv ely . Redefining X = x + λ 3 in (63), the resultan t equation coincides exactly with the integrable Eq. (31). The symmetry generators a lso coincide with the ones g iv en in Eq. (33) . So effectiv ely no new nonlinear ODE is identifie d ev en when all the t hree symmetry functions are simu ltaneously nonzero. Th us w e conclude tha t the system (1) do es not admit a three parameter Lie-p oin t sym- metry group when f ( x ) , g ( x ) 6 = 0 while the symmetry function b ( x ) = 0 in ( 8 )-(11). F urther, the only equations whic h admit t w o par a meter symmetry group alone are the four nonlinear ODEs given by Eqs. (32),(43),(48) a nd (57). IV. EQUIV ALENCE TRANSFORMA TIONS W e ha v e sho wn in t he ab ov e section tha t the iden tified ev olution equations, namely Eqs. (32), (43), (48) and (57), a dmitting t w o parameter Lie p oint symmetries can b e transformed 16 in to in tegrable equations (32), (C1), (D 1) and (E1) r esp ectiv ely , through appropria te trans- formations. In this section we give a g roup theoretical in terpretation f or these results through equiv alence transformatio ns ( ETs). W e inv oke the equiv alenc e transforma t ions and give a n explanation for the results since the group classification problem is closely related to the concept of equiv alence of equations of the ab ov e f orms with resp ect to transformations, see for example Ref. 32. Considering our original differential equation (1), let us consider a set of smo oth, lo cally one-to-one transfor ma t io ns T : ( t, x, f , g ) − → ( T , X , f 1 , g 1 ) of the space R 4 that act b y the form ulas T = F ( t, x ) , X = G ( t, x ) , f 1 = H ( t, x, f ) , g 1 = L ( t, x, g ) (66) A transformation is called an Equiv alence T ransformation (ET) of the equalit y ¨ x = − f ( x ) ˙ x − g ( x ) if it transforms the equation ¨ x = − f ( x ) ˙ x − g ( x ) (67) to an equation of the same form ¨ X = − f 1 ( X ) ˙ X − g 1 ( X ) . (68) In this case, Eqs. (67) and (68) and the f unctions { f ( x ), g ( x ) } a nd { f 1 ( X ), g 1 ( X ) } are equiv alen t 32 . It is a prov en fact that equiv alen t equations admit similar groups (for lo cal transforma- tions) and ET is a similarity transformation. That is, if (67) admits the group E then (68) also admits a gro up similar to it for lo cal transformations. Substituting the transformation (66) into Eq. (68) w e get ( G 2 t F t f 1 + G 3 t g 1 ) + ˙ x [( G 2 t F x + 2 G t F t G x ) f 1 + 3 G 2 t G x g 1 ] + ˙ x 2 [3 G 2 x G t g 1 + ( F t G 2 x + 2 G t F x G x ) f 1 ] + ˙ x 3  F x G 2 x f 1 + G 3 x g 1  = − ( G t + ˙ xG x )[( F tt + 2 ˙ xF tx + ˙ x 2 F xx − F x ( ˙ xf + g )] + ( F t + ˙ xF x )[( G tt + 2 ˙ xG tx + ˙ x 2 G xx − G x ( ˙ xf + g )] , (69) where the subscripts denote partial deriv ativ e with resp ect to that v ariable. Equating the 17 co efficien ts of differen t p ow ers of ˙ x n , n = 0 , 1 , 2 , 3, we g et F x G 2 x f 1 + G 3 x g 1 = F x G xx − G x F xx , (70) ( F t G 2 x + 2 G t F x G x ) f 1 ] + 3 G 2 x G t g 1 = F t G xx + 2 F x G tx − G t F xx − 2 G x F tx , (71) ( G 2 t F x + 2 G t F t G x ) f 1 + 3 G 2 t G x g 1 = − 2 G t F tx − G x F tt + f F x G t + 2 F t G tx − F t G x − F x G tt , (72) G 2 t F t f 1 + G 3 t g 1 = − G t F tt + g F x G t + F t G tt − g G x F t . (73) Solving Eqs. (70) and (71) consisten tly w e find G x = 0 and F xx = 0. As a result one gets G = α ( t ) , F = β ( t ) x + γ ( t ) , (74) where α, β and γ are arbitrary functions of t . Substituting Eq. (7 4 ) in (72) and (73) a nd simplifying the resultan t equations w e get ˙ α 2 β f 1 = f β ˙ α + β ¨ α − 2 ˙ α ˙ β , (75) ˙ α 2 ( ˙ β x + ˙ γ ) f 1 + ˙ α 3 g 1 = − ˙ α ( ¨ β x + ¨ γ ) + g β ˙ α + ¨ α ( ¨ β x + ¨ γ ) . (76) F rom Eq . (7 5) w e can obtain a n expression whic h connects the transformed function f 1 with the origina l function f of the form f 1 = f ˙ α + ¨ α ˙ α 2 − 2 ˙ β β ˙ α . (77) Substituting (77) in (7 6) and simplifying the resultan t equation we arriv e a t g 1 = β g ˙ α 2 − ( ˙ β x + ˙ γ ) ˙ α 2 f + 2 ˙ β ( ˙ β x + ˙ γ ) β ˙ α 2 − ( ¨ β x + ¨ γ ) ˙ α 2 . (78) Th us w e obtain the general ET T = α ( t ) , X = β ( t ) x + γ ( t ) , f 1 = f ˙ α + ¨ α ˙ α 2 − 2 ˙ β β ˙ α , g 1 = g 1 = β g ˙ α 2 − ( ˙ β x + ˙ γ ) ˙ α 2 f + 2 ˙ β ( ˙ β x + ˙ γ ) β ˙ α 2 − ( ¨ β x + ¨ γ ) ˙ α 2 . (79) Since we hav e already iden tified only four equations (vide Eqs. (32 ), (43), (48) and (57 )) that a r e inv arian t under tw o parameter Lie p oin t symmetries within the class of equations (1) w e consider only these four equations and presen t our result. Now solving Eqs. (79) 18 with the giv en fo rm of f and g one obtains the following result Case 1 (Eq.(32)) α = − ( q + 2) q k 2 e − qk 2 ( q +2) t, β = e − k 2 ( q +2) t, γ = 0 so that f 1 = α X q g 1 = β X 2 q +1 Case 2 (Eq.(43)) α = t, β = e − λ 2 2 R log [ x ( t )] dt , γ = 0 so that f 1 = c onstant g 1 = X Case 3 (Eq.(48)) α = 1 A e At , β = e At , γ = 0 so that f 1 = 1 X 2 g 1 = 1 X 3 Case 4 (Eq.(57)) α = − λ 2 λ 1 e − λ 1 λ 2 , β = 1 , γ = − λ 1 λ 2 2 t so that f 1 = e λ 2 U g 1 = e − 2 λ 2 U (80) It directly follows that with the ab ov e form of f 1 and g 1 , Eq. (68) tak es the form of (3 2), (C1), (D 1) and (E1)) resp ectiv ely , whic h w ere shown to b e in tegrable. V. LIE SYMMET RIE S OF EQ. (1) WI TH f ( x ) = 0 OR g ( x ) = 0 Next w e consider the sp ecial case of Eq. (1 ) with f ( x ) = 0 , that is, ¨ x + g ( x ) = 0 . (81) In t he following w e fo cuss our attention o nly on the case b = 0 so that w e ha v e ξ = a ( t ) , η = c ( t ) x + d ( t ). Eqs. (10) a nd (11) with b ( x ) = 0 and f ( x ) = 0 give r ise to the follo wing conditions, resp ectiv ely , ¨ a − 2 ˙ c = 0 , ( 8 2) and g x +  2 ˙ a − c cx + d  g + ¨ cx + ¨ d cx + d = 0 . (83) Since g ( x ) should b e a function of x alone, we c ho ose 2 ˙ a c − 1 = λ 1 , d c = λ 2 , ¨ c c = − λ 3 , (84) where λ 1 , λ 2 and λ 3 are constant para meters. Note that the ab ov e implies ¨ d c = λ 2 ¨ c c = λ 2 λ 3 . 19 Solving (84) w e find that t he solution exists either fo r the parametric c hoice λ 1 6 = 3 or λ 1 = 3. The resp ectiv e infinitesimal symmetries are ξ = a ( t ) = a 1 + a 2 t, η = c ( t ) x + d ( t ) = 2 a 2 x (1 + λ 1 ) + 2 λ 2 a 2 (1 + λ 1 ) , λ 1 6 = 3 (85) ξ = a ( t ) = a 1 − (1 + λ 1 ) 2 √ λ 3 ( c 2 cos p λ 3 t − c 1 sin p λ 3 t ) , η = c ( t ) x + d ( t ) = a 1 − ( c 1 cos p λ 3 t + c 2 sin p λ 3 t )( x + λ 2 ) , λ 1 = 3 (86) The resp ectiv e inv ar ian t equations turn out to b e ¨ x + g 1 ( x + λ 2 ) λ 1 = 0 , λ 1 6 = 3 , (87) and ¨ x + λ 3 4 ( x + λ 2 ) + g 1 ( x + λ 2 ) 3 = 0 , λ 1 = 3 . (88) Th us Eq. (87) admits a tw o parameter symmetry gro up with the generators X 1 = ∂ ∂ t , X 2 = t ∂ ∂ t + 2( x + λ 2 ) (1 + λ 1 ) ∂ ∂ x . (89) On the other hand Eq. (88) admits a three parameter symmetry gro up with the symmetry generators X 1 = ∂ ∂ t , X 2 = sin p λ 3 t  (1 + λ 1 ) 2 √ λ 3 ∂ ∂ t + ( x + λ 2 ) ∂ ∂ x  , X 3 = cos p λ 3 t  (1 + λ 1 ) 2 √ λ 3 ∂ ∂ t + ( x + λ 2 ) ∂ ∂ x  . (90) Redefining x + λ 2 = X in Eqs. (87) and (88) w e g et ¨ X + g 1 X λ 1 = 0 , λ 1 6 = 3 . (91) ¨ X + ω 2 X − ˜ g X 3 = 0 , ω 2 = λ 3 4 , ˜ g = − g 1 , λ 1 = 3 , (92) Equation in (91) corresp onds to a conserv ative Hamiltonian system ( H = p 2 2 + g 1 (1 − λ 1 ) X 1 − λ 1 ) and so the Liouville in tegrabilit y is assured. On the o ther hand Eq. (92) is nothing but the Pinney-Ermak o v equation, whose origin, prop erties and the method of finding its general solution hav e b een discusse d widely in the con tempo rary nonlinear dynamics literature (see for example R ef. 33 and references therein). F or the sak e of completeness w e g iv e the g eneral solution of this equation as X = 1 Aω q ( ω 2 A 4 − ˜ g ) sin 2 ( ω t + φ ) + ˜ g , (93) 20 It has also b een show n t hat Eq. ( 92) can b e transformed to harmonic oscillator equation through suitable nonlo cal transformatio n and from the solution of the la t t er one can con- struct the solution fo r the nonlinear equation. F or mor e details o ne may refer 41 . Finally , fo r g ( x ) = 0, Eq. (1 ) can b e written as ¨ x + f ( x ) ˙ x = 0 . (94) Eq. (94) can b e transformed to a first order equation b y a trivial change of v aria ble which in turn can also b e in tegrated trivially . So w e do not discuss the symm etries of this equation here. VI. CONCLUSIO NS In the prese n t pap er w e hav e in v estigated the Li ´ enard t yp e equation (1) in the framew ork of mo dern g roup analysis of differen tial equations. Ev en tho ug h the in tegrabilit y prop erties of some of the sp ecific equations coming under the Li´ enard type ha v e b een discussed in the literature, w e ha v e identified all those equations whic h a dmit only t w o and three para meter symmetry gr oups. T o identify the in tegrable equations b elonging to the class (1) we ha v e deduced all the equations that are inv a r ian t under one, t w o and three par a meter Lie p oin t symmetries. Ob- viously the general Eq. (1) do es not contain the v ariable t explicitly and so it alw a ys admits a time translationa l generator. How eve r, w e hav e demonstrated that sev eral equations admit t w o parameter Lie p oin t symmetry groups. In pa rticular these equations corresp ond to four sp ecific forms of the f unctions f ( x ) a nd g ( x ) in (1), see Eq. (2), namely Eqs. (3 2), ( 43), (48) and (57). These equations ha v e b een deduced here through a group theoretical p oint of view alone. W e hav e also discusse d the in tegrability prop erties of these equations briefly and sho wn the existence of equiv alenc e transformations. After analyzing the Lie p oint sym- metries w e hav e also sho wn that Li´ enard type equation do es not admit a thr ee parameter symmetry group when b oth f ( x ) , g ( x ) 6 = 0 in Eq. (1). How eve r, in the sub-case, f ( x ) = 0, one can find that the w ell kno wn Pinney-Ermak o v equation is the only equation whic h is in v ariant under a three pa r a meter Lie p oin t symmetry gro up. In this pap er w e ha v e restricted our atten tion only on the non-maximal Lie p oin t symme- try groups. The question whic h nat ura lly a rises is what happ ens if one considers the more 21 general case, b 6 = 0, vide Eq. (17). Suc h an analysis allows us to isolate a class of equations admitting eight parameter symmetries. W e will presen t the results in t he follow -up pap er I I. Ac kno wledgmen ts One of us (SNP) is gr ateful to the Cen tre for Nonlinear D ynamics, Bharathida san Univ er- sit y , Tiruc hirappalli, for w arm hospitalit y . The w ork of SNP forms part of a Department of Science a nd T ec hnology , Go v ernmen t o f India sp onsored researc h pro ject. The work of MS forms part of a researc h pro ject sp o nsored b y Nat io nal Board f o r Higher Mathematics, Gov- ernmen t of India. The w ork of ML forms part of a D epartmen t of Science and T echnology (DST), R a manna F ellowsh ip and is also supp or t ed by a D ST-IRHP A researc h pro ject. In the follo wing, w e briefly discuss the inte grability prop erties of the equations deriv ed in Sec. I I I. T o b egin with let us consider the Liouville in tegrabilit y of Eqs. (33). APPENDIX A: TIME INDEPEN DE NT HAMI L TONIAN FOR ( 37) Recen tly , w e hav e studied the in tegrabilit y of (32 ) or equiv alen tly (3 7) a nd found that it admits time indep enden t integrals fo r all v alues of the parameters α and β 22,34 . F rom the time indep enden t in tegrals w e hav e iden tified the follow ing time indep enden t Hamilto nia n for (37), namely , H =          ( r − 1) ( r − 2) p ( r − 2) ( r − 1) − ( r − 1) r ˆ αpw q +1 , α 2 > 4 β ( q + 1) ˆ α 2 pw q +1 + log ( 1 p ) , α 2 = 4 β ( q + 1) 1 2 log h w 2( q +1) ( q + 1) 2 sec 2 [ ω ( q + 1) w q +1 p ] i − ˆ α 2 pw q +1 , α 2 < 4 β ( q + 1) , (A1) where the corresp onding canonically conjug ate momen tum is defined by p =     ˙ w + ( r − 1) r ˆ αw q +1  (1 − r ) , α 2 ≥ 4 β ( q + 1 ) ( q + 1) ω w q +1 tan − 1 h αw q +1 +2( q + 1) ˙ w 2 ωw q +1 i α 2 < 4 β ( q + 1) , (A2) where r = α 2 β ( q +1) ( α ± p α 2 − 4 β ( q + 1 ), ω = 1 2 p 4 β ( q + 1) − α 2 and ˆ α = α q +1 . F or more details a b out the deriv atio n of the ab o v e Hamiltonian one ma y refer to Ref. 34. The time indep enden t Hamiltonian ensures the L io uville integrabilit y of (3 2) or (37 ). 22 APPENDIX B: NOT ABLE INT E GRABLE E QUA TIONS I N (32) Besides the general case, q = ar bitr ar y , Eq. (32) encompasses sev eral know n integrable equations of contem p orary intere st. The inte resting equations can b e identifie d by a ppro- priately c ho osing the parameter q as w e demonstrate briefly in the following. F or example, c ho osing q = 1 in (32) one g ets the g eneralized MEE, ¨ x + ( k 2 + f 1 x ) ˙ x + 2 k 2 2 9 x + k 2 f 1 3 x 2 + g 1 x 3 = 0 , (B1) Eq. (B1) can b e transformed into the MEE, w ′′ + f 1 w w ′ + g 1 w 3 = 0, b y intro ducing a transformation w = xe k 2 3 t and z = − 3 k 2 e − k 2 3 t . The Hamiltonian structure fo r t his equation can b e extracted fro m (A1) b y restricting q = 1 in the latter relations. The restriction f 1 = 0 in (B1) provide s us the force-free Duffing oscillator whose in v a r iance and in tegrabilit y prop erties ha v e b een discuss ed in R efs.14,35. With the c hoice f 1 = 3 , g 1 = 1 , k 2 = 0 , the resultan t equation b ecomes a linearizable one whose inv ariance and in tegrability prop erties ha v e b een discussed in detail in Refs. 36,37,38,39. The case q = 2 in (32) g iv es us ¨ x + ( k 2 + f 1 x 2 ) ˙ x + 3 k 2 2 16 x + k 2 f 1 4 x 3 + g 1 x 5 = 0 . (B2) The explicit form o f the Ha miltonian can be fixed from (A1 ) by restricting q = 2 in the latter relations. W e no t e here tha t Eq. (B2) also includes sev eral know n integrable equations. The notable example s are force-free Duffing-v an der P ol oscillator equation ( g 1 = 0) and the second equation in t he MEE hierarc h y ( f 1 = 0 , g 1 = 1 16 ). Finally , w e note that one may also reco v er specific equations lik e the force-free Helmholtz oscillator and the asso ciated Lie symmetries can b e obtained by appropriately c ho osing the v alue of the parameter q . Cho osing q = 1 2 and f 1 = 0 in (32) one gets t he force-free Helmhotz oscillator. The symmetries of (3 3) with q = 1 2 coincide exactly with the one rep orted in Ref. 40. APPENDIX C: METHOD OF INTEGRA TING EQ . (43) The solution o f Eq. (43) can b e constructed from the solution of the damp ed harmonic oscillator using a general pro cedure giv en b y us sometime ago in Ref. 41. F or example, let 23 us consider a linear ODE of the form ¨ U + α ˙ U + g 1 U = 0 , (C1) where α and g 1 are arbitrary parameters. By in tro ducing a nonlo cal transformation of the form U = xe λ 2 2 R t log x ( t ′ ) dt ′ in the linear O D E (C1) the latter can b e brough t to the form ¨ x + ( α + λ 2 2 + λ 2 log( x )) ˙ x + g 1 x + αλ 2 2 x log x + λ 2 2 4 x (log x ) 2 = 0 . (C2) No w redefining the constants ( α + λ 2 2 ) = − f 1 in (C2) o ne exactly ends up with (4 3). F ollowing the pro cedure give n in Ref. 41 o ne can obt a in the general solution for (C2) from the linear equation. APPENDIX D: METHOD OF INTEGRA TING E Q . (48) Eq. (4 8) can b e transformed to the equation of the form ¨ U + f 1 U 2 ˙ U + g 1 U 3 = 0 , ( D 1) through t he tra nsformation U = xe A 2 t , Z = 1 A e At . Eq. (D1) admits Hamiltonian structure for all v alues of f 1 and g 1 . The underlying Hamiltonian reads H =          ( r − 1) ( r − 2) p ( r − 2) ( r − 1) + ( r − 1) f 1 r p U , f 2 1 > − 4 g 1 log( 1 p ) − f 1 2 p U , f 2 1 = − 4 g 1 f 1 2 p U + 1 2 log  1 U 2 sec 2 [ − ω p U  , f 2 1 < − 4 g 1 , (D2) where the canonical conjug ate momen tum is defined b y p =    1 ( r − 1) ( ˙ U − ( r − 1) f 1 r U ) (1 − r ) , f 2 1 ≥ 4 g 1 − U ω tan − 1 [ f 1 − 2 U ˙ U 2 ω ] , f 2 1 < 4 g 1 (D3) where r = − f 1 2 g 1 ( f 1 ± p f 2 1 + 4 g 1 ) and ω = 1 2 p − 4 g 1 − f 2 1 . The time indep enden t Hamiltonian giv en ab ov e ensures the Liouville in tegrabilit y of Eq. (D1). APPENDIX E: ME T HOD OF I NTEGRA TING EQ. (57) By in tro ducing a tra nsformation U = x − λ 1 λ 2 2 t and z = − λ 2 λ 1 e − λ 1 λ 2 t in (57) the latter can b e transformed into the for m U ′′ + f 1 e λ 2 U U ′ + g 1 e − 2 λ 2 U = 0 . ( ′ = d dz ) (E1) 24 Eq. (E1) can b e rewritten in t he form fo rm U ′′ + f 1 f ( U ) U ′ + ˜ g 1 f ( U ) Z f ( U ) dU = 0 , (E2) where f ( U ) = e − λ 2 U and ˜ g 1 = − λ 2 g 1 . Eq. (E2) a dmits time indep enden t Hamiltonian fo r all v alues of f 1 and g 1 . The resp ectiv e Hamiltonians a re H =          ( r − 1) ( r − 2) p ( r − 2) ( r − 1) + ( r − 1) f 1 r λ 2 pe − λ 2 U , f 2 1 > 4 ˜ g 1 log[ p ] + f 1 p 2 λ 2 e − λ 2 U , f 1 = 4 ˜ g 1 1 2 log h e − 2 λ 2 U λ 2 2 sec 2 [ ω pe − λ 2 U − λ 2 ] i + f 1 2 λ 2 pe − λ 2 U , f 1 < 4 ˜ g 1 , (E3) where the canonically conjugate momen tum is defined b y p =    [ U ′ + (1 − r ) r f 1 e − λ 2 U λ 2 ] 1 − r , f 2 1 ≥ 4 ˜ g 1 − λ 2 e λ 2 U ω tan − 1 [ 2 λ 2 U ′ − f 1 e − λ 2 U 2 ωe − λ 2 U ] , f 2 1 < 4 ˜ g 1 (E4) where r = f 1 2 g 1 ( f 1 ± p f 2 1 − 4 ˜ g 1 ) , ω = 1 2 p 4 ˜ g 1 − f 2 1 . One ma y note that the a b o v e Hamiltonian resem bles t he Hamiltonian structure of (A1). 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