  Exact solutions are obtained in the quadratic theory of gravity with a scalar
field for wave-like models of space-time with spatial homogeneity symmetry and
allowing the integration of the equations of motion of test particles in the
Hamilton-Jacobi formalism by the method of separation of variables with
separation of wave variables (Shapovalov spaces of type II). The form of the
scalar field and the scalar field functions included in the Lagrangian of the
theory is found. The obtained exact solutions can describe the primary
gravitational wave disturbances in the Universe (primary gravitational waves).
