Caratheodory periodic perturbations of degenerate systems

We study the structure of the set of harmonic solutions to T-periodically  perturbed coupled differential equations on differentiable manifolds, where the  perturbation is allowed to be of Caratheodory-type regularity.  Employing degree-theoretic methods, we prove the existence of a noncompact connected  set of nontrivial T-periodic solutions that, in a sense, emanates from the set of zeros of the unperturbed vector field. The latter is assumed to be ''degenerate'': Meaning that, contrary to the usual assumptions on the leading vector field,  it is not required to be either trivial nor to have a compact set of zeros.  In fact, known results in the ``nondegenerate case can be recovered from our ones.  We also provide some illustrating examples of Lienard- and \(\phi\)-Laplacian-type  perturbed equations. For more information see https://ejde.math.txstate.edu/Volumes/2024/39/abstr.html