  The closed string model in the background gravity field is considered as the
bi-Hamiltonian system in assumption that string model is the integrable model
for particular kind of the background fields. The dual nonlocal Poisson
brackets, de pending of the background fields and of there derivatives, are
obtained. The integrability condition is formulated as compatibility of the
bi-Hamiltonity con dition and the Jacobi identity of the dual Poisson bracket.
Two types of the non local brackets are introduced. Constant curvature and
time-dependent metrics are considered, as example. It is, shown that the Jacobi
identities for the non local brackets have particular solution for the
space-time coordinates, as matrix representation of the simple Lie group.
