  In this work, two particular orthogonal and conformal decompositions of the
3+1 dimensional Einstein equation and Arnowitt-Deser-Misner (ADM) formalism for
general relativity are obtained. In order to do these, the 3+1 foliation of the
four-dimensional spacetime, the fundamental conformal transformations and the
Hamiltonian form of general relativity that leads to the ADM formalism, defined
for the conserved quantities of the hypersurfaces of the globally-hyperbolic
asymptotically flat spacetimes, are reconstructed. All the calculations up to
chapter 7 are just a review.
  We propose a method in chapter 7 which gives an interesting relation between
the Cotton (Conformal) soliton and the static vacuum solutions. The formulation
that we introduce can be extended to find the gradient Cotton soliton and the
solutions of Topologically Massive Gravity (TMG) as well as the gradient Ricci
soliton.
