Tangential center problem for planar Hamiltonian differential equations

Consider a planar Hamiltonian differential equation with a nondegenerate center at the origin. The Tangential Center Problem asks for a characterization of polynomial perturbations for which the center persists at first order. This problem plays a fundamental role in the study of the center-focus problem. We propose an effective algorithmic approach to the Tangential Center Problem for planar polynomial Hamiltonian differential equations. Using this approach, we obtain a complete solution of the quadratic Tangential Center Problem. We further apply it to a non-generic family of cubic Hamiltonian differential equations containing the planar Hamiltonian Lotka--Volterra equation. For a general cubic polynomial perturbation, we derive necessary and sufficient conditions for the vanishing of the corresponding first Poincaré--Pontryagin--Melnikov function and characterize all tangential centers.