Invariant surfaces of standard two-dimensional systems with conservative first approximation of the third order

We simultaneously study two classes of two-dimensional time-periodic systems of differential equations with a small positive parameter, namely, systems with “slow” or “fast” time whose first-approximation systems are autonomous and conservative and do not contain terms of order higher than three. Thus, the corresponding unperturbed systems have one, two, or three rest points. For the perturbations, we indicate explicit conditions, independent of the small parameter, under which every original system of either class with coefficients three times continuously differentiable with respect to the phase variables and the parameter in a neighborhood of zero has finitely many two-dimensional invariant surfaces homeomorphic to tori for all sufficiently small parameter values. We also give formulas for these surfaces.