In this paper we study a pair V₁,V₂ of commuting isometric operators in a Hilbert space H, which are not necessarily defined on the whole space. An old question: is there a possibility for an extension of V₁,V₂ to a pair of commuting unitary operators U₁,U₂ in a Hilbert space \widetilde H⊇ H? In the case of a unitary V₂ we present a transparent criterion in terms of the original space H. The general case is discussed, as well. We introduce a notion of a generalized resolvent for V₁,V₂. Characteristic properties of the generalized resolvent in terms of H are obtained. In the case of a unitary V₂ all generalized resolvents are parametrized.