  The closed string model in the background gravity field and the antisymmetric
B-field is considered as the bihamiltonian system in assumption,that string
model is the integrable model for particular kind of the background fields. It
is shown, that bihamiltonity is origin of two types of the T-duality of the
closed string models. The dual nonlocal Poisson brackets, depending of the
background fields and of their derivatives, are obtained. The integrability
condition is formulated as the compatibility of the bihamoltonity condition and
the Jacobi identity of the dual Poisson bracket. It is shown, that the dual
brackets and dual hamiltonians can be obtained from the canonical (PB) and from
the initial hamiltonian by imposing of the second kind constraints on the
initial dynamical system, on the closed string model in the constant background
fields, as example. The closed string model in the constant background fields
is considered without constraints, with the second kind constraints and with
first kind constraints as the B-chiral string. The two particles discrete
closed string model is considered as two relativistic particle system to show
the difference between the Gupta-Bleuler method of the quantization with the
first kind constraints and the quantization of the Dirac bracket with the
second kind constraints.
