We deal with nonlinear periodic differential systems depending on a small parameter. The unperturbed system has an invariant manifold of periodic solutions. We provide sufficient conditions in order that some of the periodic orbits of this invariant manifold persist after the perturbation. These conditions are not difficult to check, as we show in some applications. The key tool for proving the main result is the Lyapunov--Schmidt reduction method applied to the Poincaré--Andronov mapping.

This paper deals with the notion of integrability of flows or vector fields on two-dimensional manifolds. We consider the following two key points about first integrals: (1) They must be defined on the whole domain of definition of the flow or vector field, or defined on the complement of some special orbits of the system; (2) How are they computed? We prove that every local flow ϕ on a two-dimensional manifold M always has a continuous first integral on each component of M\Σ where Σ is the set of all separatrices of ϕ. We consider the inverse integrating factor and we show that it is better to work with it instead of working directly with a first integral or an integrating factor for studying the integrability of a given two-dimensional flow or vector field. Finally, we prove the existence and uniqueness of an analytic inverse integrating factor in a neighborhood of a strong focus, of a non-resonant hyperbolic node, and of a Siegel hyperbolic saddle.

Let X(x,y) and Y(x,y) be real analytic functions without constant and linear terms defined in a neighborhood of the origin. Assume that the analytic differential system \dot{x}=y+ X(x,y), \dot{y}=Y(x,y) has a nilpotent center at the origin. The first integrals, formal or analytic, will be real except if we say explicitly the converse. We prove the following. If X= y f(x,y^2) and Y= g(x,y^2), then the systemhas a local analytic first integral of the form H=y^2+F(x,y),where F starts with terms of order higher than two.If the system has a formal first integral, then it hasa formal first integral of the form H=y^2+F(x,y), where Fstarts with terms of order higher than two. In particular, if thesystem has a local analytic first integral defined at the origin,then it has a local analytic first integral of the formH=y^2+F(x,y), where F starts with terms of order higher than two.As an application we characterize the nilpotent centersfor the differential systems \dot{x}=y+P_3(x,y),\dot{y}=Q_3(x,y), which have a local analytic first integral,where P_3 and Q_3 are homogeneous polynomials of degree three.