  For the quadratic theory of gravity with a scalar field, exact solutions are
found for gravitational-wave models in Shapovalov~I~type spacetimes, which do
not arise in models of the general theory of relativity. The theory of gravity
under consideration can effectively describe the early stages of the dynamics
of the universe. Type I Shapovalov spaces are the most general form of
gravitational-wave Shapovalov spacetimes, whose metric in privileged coordinate
systems depends on three variables, including the wave variable. For Einstein
vacuum spacetimes, these wave models degenerate into simpler types. The
obtained exact models of gravitational waves in the quadratic theory of gravity
can be used to test the realism of such theories of gravity.
