Singular symplectic surfaces

In this paper we classify all singular irreducible symplectic surfaces, i.e., compact, connected complex surfaces with canonical singularities that have a holomorphic symplectic form σ on the smooth locus, and for which every finite quasi-\'etale covering has the algebra of reflexive forms spanned by the reflexive pull-back of σ. We moreover prove that the Hilbert scheme of two points on such a surface X is an irreducible symplectic variety, at least in the case where the smooth locus of X is simply connected.