Stability Conditions For Reversible And Partially Integrable Systems

The purpose of this paper is to provide results like Peixoto's Theorem inside a class X of 3-dimensional reversible systems. We deal with reversible systems associated to an involution phi(1) such that Dim Fix(phi(1)) is equal to one. We further assume the existence of a first integral for any X is an element of X that leaves invariant any sphere centered at the origin. The main result establishes generic necessary conditions for the structural stability in X. Stability conditions on generic one-parameter families of reversible vector fields on the two-dimensional sphere are also stated.