Deducing properties of ODEs from their discretization

In this paper we address the question Q1: What rational conserved integral(s) and (inverse) polynomial volume form (if any) does the ODE possess? Since finding rational integrals generally requires solving a nonlinear problem we propose a three step program, that, using a certain ansatz, only requires the solution of linear problems: Step 1: Discretise the ODE using a “suitable” method. In this paper we will use Kahan’s method (but much of the following also holds for certain other birational integration methods given in the references). Compute the Jacobian determinant J of the discretisation, and factorise J . Step 2: Use the factors of J as candidate discrete cofactors for finding discrete Darboux polynomials (DPs). Step 3: Take the continuum limits of the discrete cofactors and DPs found in step 2. If possible, use these DPs as building blocks for time-dependent/time-independent first integrals and preserved measure of the ODE, If one is very lucky, it may even be possible to use them to derive the exact solution to the initial value problem fore the ODE.