On the finite cyclicity of open period annuli

Let $\mathrm{\Pi }$ be an open, relatively compact period annulus of real analytic vector field $X_0$ on an analytic surface. We prove that the maximal number of limit cycles which bifurcate from $\mathrm{\Pi }$ under a given multi-parameter analytic deformation $X_{\lambda }$ of $X_0$ is finite, provided that $X_0$ is either Hamiltonian, or generic Darbouxian vector field.