  The closed string model in the background gravity field is considered as a
bi-Hamiltonian system in assumption that string model is the integrable model
for particular kind of the background fields. The dual nonlocal Poisson
brackets(PB), depending of the background fields and of their derivatives, are
obtained. The integrability condition is formulated as the compatibility of the
bi-Hamiltonity condition and the Jacobi identity of the dual PB. It is shown
that the dual brackets and dual Hamiltonians can be obtained from the canonical
PB and from the initial Hamiltonian by imposing the second kind constraints on
the initial dynamical system, on the closed string model in the constant
background fields, as example. The hydrodynamical type equation was obtained.
Two types of the nonlocal brackets are introduced. Constant curvature and
time-dependent metrics are considered. It is shown that the Jacobi identities
for the nonlocal brackets have particular solution for the space-time
coordinates, as matrix representation of the simple Lie group.
