  Let $\Pi$ 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 $\Pi$ 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.
