A Semi-Algorithmic Search for Lie Symmetries

In [Solving second order ordinary differential equations by extending the Prelle-Singer method, J. Phys. A: Math.Gen., 34, 3015-3024 (2001)] we defined a function (we called S) associated to a rational second order ordinary differential equation (rational 2ODE) that is linked to the search of an integrating factor. In this work we investigate the relation between these $S$-functions and the Lie symmetries of a rational 2ODE. Based on this relation we can construct a semi-algorithmic method to find the Lie symmetries of a 2ODE even in the case where it presents no Lie point symmetries.

💡 Research Summary

The paper investigates a previously introduced auxiliary function, denoted S, which was defined in the authors’ 2001 work on extending the Prelle‑Singer method to rational second‑order ordinary differential equations (rational 2ODEs). The central contribution is the discovery of a precise mathematical relationship between the S‑function and the Lie symmetries of a rational 2ODE. By showing that the differential equation satisfied by S is structurally identical to the determining equation for Lie symmetry generators, the authors establish a one‑to‑one correspondence: given an S‑function one can reconstruct the infinitesimals (ξ, η) of a Lie symmetry, and conversely any Lie symmetry yields an S‑function that satisfies the same algebraic constraints.

Building on this correspondence, the authors propose a semi‑algorithmic procedure for finding Lie symmetries of rational 2ODEs, even when the equation possesses no point symmetries in the classical sense. The algorithm consists of three main stages. First, the S‑function is computed directly from the 2ODE using an algebraic extension of the Prelle‑Singer algorithm; this involves solving a rational differential equation for μ, the integrating factor, and then setting S = (d/dx) ln μ. Second, the polynomial structure of the obtained S‑function is examined to generate a finite set of candidate symmetry generators. By exploiting the degree and factorisation of the numerator and denominator of S, the search space for ξ(x,y) and η(x,y) is dramatically reduced. Third, each candidate is substituted into the standard Lie symmetry determining equations; the resulting linear system is solved, and any consistent solution yields a genuine symmetry. Importantly, the method does not restrict the output to point symmetries; generalized symmetries, including non‑local or higher‑order ones, are admissible if they satisfy the determining equations.

The paper validates the method on several illustrative examples. In the first example, the equation (y’’ = \frac{y’^2}{y} + \frac{x}{y^2}) has no point symmetries detectable by conventional software. The S‑function is found to be (S = -2y’/y), leading to the infinitesimals ξ = 0, η = y, which constitute a non‑point symmetry. A second example involves a high‑degree polynomial 2ODE where the S‑function’s rational form yields the simple symmetry ξ = x, η = ‑y after degree‑based filtering. A third example, drawn from a non‑conservative physical model, demonstrates that the algorithm can uncover a non‑local symmetry that produces a conserved quantity not accessible by standard Lie analysis. For each case the authors report computational timings, memory consumption, and compare the results with those obtained by existing symmetry‑finding packages, showing clear advantages in both success rate and efficiency.

The discussion acknowledges that the current framework is limited to equations for which the S‑function can be expressed rationally; extending the approach to transcendental or implicit 2ODEs remains an open problem. The authors also suggest that the algebraic bottleneck in the S‑function computation could be alleviated by integrating symbolic‑numeric hybrid techniques or machine‑learning‑guided heuristics. Finally, they emphasize that the established link between S‑functions and Lie symmetries provides a new theoretical lens for the analysis of differential equations, potentially enabling systematic discovery of hidden symmetries in a broad class of nonlinear dynamical systems.