The center problem for discontinuous Liénard differential equation

We prove that the Lyapunov constants for differential equations given by a vector field with a line of discontinuities are quasi-homogeneous polynomials. This property is strongly used to derive the general expression of the Lyapunov constants for two families of discontinuous Liénard differential equations, modulus some unknown coefficients. In one of the families these coefficients are determined and the center problem is solved. In the other family the center problem is just solved by assuming that the coefficients which appear in these expressions are nonzero. This assumption on the coefficients is supported by their computation (analytical and numerical) for several cases.