The 1:-2 resonant center problem in the quadratic case is to find necessary and sufficient conditions (on the coefficients) for the existence of a local analytic first integral for the vector field $(x+A_1{x}^2+B_{1}xy+C{y}^2){\partial }_x+(-2y+D{x}^2+A_{2}xy+B_2{y}^2){\partial }_y$. There are twenty cases of center. Their necessity was proved in [4] using factorization of polynomials with integer coefficients modulo prime numbers. Here we show that, in each of the twenty cases found in [4], there is an analytic first integral. We develop a new method of investigation of analytic...

A new concept of stability, closely related to that of structural stability, is introduced and applied to the study of C¹ endomorphisms with singularities. A map that is stable in this sense is conjugate to each perturbation that is equivalent to it in a geometric sense. It is shown that this kind of stability implies Axiom A and Ω-stability, and that every critical point is wandering. A partial converse is also shown, providing new examples of C³ structurally stable maps.