Abstract. We consider the symmetry properties of an integro-differential multidimensional Gross-Pitaevskii equation with a nonlocal nonlinear (cubic) term in the context of symmetry analysis using the formalism of semiclassical asymptotics. This yields a semiclassically reduced nonlocal Gross-Pitaevskii equation, which can be treated as a nearly linear equation, to determine the principal term of the semiclassical asymptotic solution. Our main result is an approach which allows one to construct a class of symmetry operators for the reduced Gross-Pitaevskii equation. These symmetry operators are determined by linear relations including intertwining operators and additional algebraic conditions. The basic ideas are illustrated with a 1D reduced Gross-Pitaevskii equation. The symmetry operators are found explicitly, and the corresponding families of exact solutions are obtained.

For a nonlocal Gross-Pitaevskii operator, quasi-energy spectral series have been constructed accurate to O( 3/2 ).These series correspond to the closed phase trajectories of the Hamilton-Ehrenfest system, which are stable in a linear approximation. The corresponding monodromy operator has been constructed accurate toin the class of trajectory-coherent functions.

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