It is possible to describe the gravitational field via a theory of 4d
surface embedded into the 10d flat space. For the first time such a
model, in which the usual form of action written down in terms of
embedding function is used, was considered by T.Regge, C.Teitelboim
in 1975. However in such approach the equations of the gravitational
field are more general than Einstein's equations, i.e. the theory is
not equivalent to General Relativity. To achieve the equivalence it is
proposed to modify the theory by additional imposition of constraints
under the construction of canonical formalism. Finally the theory with
eight-parametric gauge symmetry is obtained. This theory turns out to
be equivalent to Einstein's General Relativity if we partially fix the
gauge or redefine the metric taking for it an expression which is
invariant under additional gauge transformations. The action for such
a theory is constructed.