Discretization of Hamiltonian systems and intersection theory

We discuss the possibility of using the intersection points of the common level surface of integrals of motion with an auxiliary curve to construct finite-difference equations corresponding to different discretizations of the original integrable system. As an example, we consider the generalized one-dimensional oscillator with third- and fifth-degree nonlinearity, for which we show that the intersection divisors of the hyperelliptic curve with straight lines, quadrics, and cubics generate families of integrable discrete maps.