  This research is an extension of the author's article \cite{zar}, in which
conformally invariant generalization of string theory was suggested to
higher-dimensional objects. Special cases of the proposed theory are Einstein's
theory of gravity and string theory. This work is devoted to the formation of
self-consistent equations of the theory of induced gravity \cite{zar},
\cite{zari} in the presence of matter in the form of a perfect fluid that
interacts with scalar fields. The study is done to solve these equations for
the case of the cosmological model. In this model time-evolving gravitational
and cosmological "constants" take place which are determined by the square of
scalar fields. The values of which can be matched with the observational data.
The equations that describe the theory have solutions that can both match with
the solutions of the standard theory of gravity as well as it can differ from
it. This is due to the fact that the fundamental "constants" of the theory,
such as gravitational and cosmological, can evolve over time and also depend of
the coordinates. Thus, in a rather general case the theory describes the two
systems (stages): Einstein and "evolving" or "restructuring" (the name
suggested by the author). This process is similar to the phenomenon of phase
transition, where the different phases (Einstein gravity system, but with
different constants) transit into each other.
