  Let G be a (real or complex) linear reductive algebraic group acting on an
affine variety V. Let W be a subvariety. In this work we study how the G-orbits
intersect W.
  We develop a criterion to determine when the intersection can be described as
a finite union of orbits of a reductive subgroup. The conditions of the
criterion are easily verified in practice and are used to construct continuous
families of (non-isomorphic) nilpotent Lie groups which do not admit
left-invariant Ricci soliton metrics. Other applications to the left-invariant
geometry of Lie groups are also given.
  The note finishes by applying our techniques to the adjoint representation.
The classical result of finiteness of nilpotent orbits is reproven and it is
shown that each of these orbits contains a critical point of the norm squared
of the moment map.
