  S. Ovsienko proved that the Gelfand-Tsetlin variety for $\mathfrak{gl}_n$ is
equidimensional (i.e. all its irreducible components have the same dimension)
with dimension equals $\frac{n(n-1)}{2}$. This result has important
consequences in Representation Theory of Algebras, implying, in particular, the
equidimensionality of the nilfiber of the Kostant-Wallach map. In this paper we
will present the generalization of this result and will address a weak version
of Ovsienko's Theorem which includes the regular case.
