On the behavior of orbits of Vanhaecke system on integral surfaces

In the 1990s, P. Vanhecke described a Hamiltonian system with two degrees of freedom and a polynomial Hamiltonian integrable in Abelian functions of two variables. This system provides a convenient example of an integrable system in which integral curves are wound on a two-dimensional manifold, an algebraic surface in a 4-dimensional phase space. In this report, we show that all necessary calculations can be performed in the Sage system. The role of periods of Abelian integrals and their commensurability in describing the nature of the winding of integral curves on an algebraic integral surface is discussed. The results of numerical experiments performed in fdm for Sage are presented.