Two classes of time-periodic systems of ordinary differential equations with a small parameter ε ≥ 0, those with “fast” and “slow” time, are studied. The corresponding conservative unperturbed systems (Formula presented.) have 1 to 3ⁿ singular points. The following results are obtained in explicit form: (1) conditions on perturbations independent of the parameter under which the initial systems have a certain number of invariant surfaces of dimension n + 1 homeomorphic to the torus for all sufficiently small parameter values; (2) formulas for these surfaces and their asymptotic expansions; (3) a description of families of systems with six invariant surfaces.