On algebraic construction of certain integrable and super-integrable systems

We propose a new construction of two-dimensional natural bi-Hamiltonian systems associated with a very simple Lie algebra. The presented construction allows us to distinguish three families of super-integrable monomial potentials for which one additional first integral is quadratic, and the second one can be of arbitrarily high degree with respect to the momenta. Many integrable systems with additional integrals of degree greater than two in momenta are given. Moreover, an example of a super-integrable system with first integrals of degree two, four and six in the momenta is found.

💡 Research Summary

The paper introduces a novel algebraic framework for constructing two‑dimensional natural Hamiltonian systems that are bi‑Hamiltonian and, in many cases, super‑integrable. The authors start from the standard symplectic Poisson tensor (P_{0}) and introduce a second, compatible Poisson tensor (P_{1}) derived from a very simple two‑dimensional Lie algebra (\mathfrak g=\langle e_{1},e_{2}\rangle) with the single non‑trivial bracket ({e_{1},e_{2}}=e_{2}). By requiring that the Hamiltonian \