Bihamiltonity as the origin of T-duality of the closed string model

In assumption that a string model is an integrable model for a particular kind of background fields, the closed string model in the background gravity field and in the antisymmetric B-field is considered as a bihamiltonian system. It is shown that bihamiltonity is the origin of T-duality of the string models. The new Poisson brackets, which depend on the background fields and on their derivatives, are obtained. The integrability condition is obtained as the compatibility of the bihamiltonity condition and the Jacobi identity of the new Poisson bracket. The B-chiral string model is dual to he chiral string model for constant background fields.