  $[n+1]$-dimensional ($n\geq 3$) smooth Einsteinian spaces of Euclidean and
Lorentzian signature are considered. The base manifold $M$ is supposed to be
smoothly foliated by a two-parameter family of codimension-two-surfaces which
are orientable and compact without boundary in $M$. By applying a pair of
nested $1+n$ and $1+[n-1]$ decompositions, the canonical form of the metric and
the conformal structure of the foliating codimension-two-surfaces a gauge
fixing, analogous to the one applied in arXiv:1409.4914, is introduced. In
verifying that the true degrees of freedom of gravity may conveniently be
represented by the conformal structure it is shown first that regardless
whether the primary space is Riemannian or Lorentzian, in terms of the chosen
geometrically distinguished new variables, the $1+n$ momentum constraint can be
written as a first order symmetric hyperbolic system. It is also argued that in
the generic case the Hamiltonian constraint can be solved as an algebraic
equation. By combining the $1+n$ constraints with the part of the reduced
system of the secondary $1+[n-1]$ decomposition that governs the evolution of
the conformal structure---in the Riemannian case with an additional
introduction of an imaginary `time'---a well-posed mixed
hyperbolic-algebraic-hyperbolic system is formed. It is shown that regardless
whether the primary space is of Riemannian or Lorentzian if a regular origin
exists solutions to this system are also solutions to the full set of
Einstein's equations. This, in particular, offers the possibility of developing
a new method for solving Einstein's equations in the Riemannian case. The true
degrees of freedom of gravity are also found to be subject of a nonlinear wave
equation.
