  Based on the ideology of the Maslov's complex germ theory, a method has been
developed for finding an exact solution of the Cauchy problem for a
Hartree-type equation with a quadratic potential in the class of
semiclassically concentrated functions. The nonlinear evolution operator has
been obtained in explicit form in the class of semiclassically concentrated
functions. Parametric families of symmetry operators have been found for the
Hartree-type equation. With the help of symmetry operators, families of exact
solutions of the equation have been constructed. Exact expressions are obtained
for the quasi-energies and their respective states. The Aharonov-Anandan
geometric phases are found in explicit form for the quasi-energy states.
