Commuting planar polynomial vector fields for conservative newton systems

We study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field on a plane. It is a classical result that one can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent (transversal) commuting vector field (see Theorem 2.1). In what follows, we will use the standard correspondence between (polynomial) vector fields and derivations on (polynomial) rings. Let [MATH HERE] be a derivation, where f is a polynomial with coefficients in a field K of zero characteristic. This derivation corresponds to the differential equation ẍ = f(x), which is called a conservative Newton system as it is the expression of Newton's second law for a particle confined to a line under the influence of a conservative force. Let H be the Hamiltonian polynomial for d with zero constant term, that is [MATH HERE]. Then the set of all polynomial derivations that commute with d forms a K[H]-module M_d [6, Corollary 7.1.5]. We show that, for every such d, the module M_d is of rank 1 if and only if deg f ⩾ 2. For example, the classical elliptic equation ẍ = 6x² + a, where a ∈ C, falls into this category. For proofs of the results stated in this abstract, see [5].