Simple connection between the geometric and the Hamiltonian representations of integrable nonlinear equations

One gives a simple and general derivation of the well-known connection between the geometric and the Hamiltonian approaches in the classical method of the inverse problem. Namely, for the case of a two-dimensional auxiliary problem and periodic boundary conditions it is explicitly shown how the existence of the classical $r$-matrix for the integrable equations leads to their representation in the form of the condition of zero curvature.