  In this work we investigate solvable and nilpotent Lie groups with special
metrics. The metrics of interest are left-invariant Einstein and algebraic
Ricci soliton metrics. Our main result shows that the existence of a such a
metric is intrinsic to the underlying Lie algebra. More precisely, we show how
one may determine the existence of such a metric by analyzing algebraic
properties of the Lie algebra in question and infinitesimal deformations of any
initial metric.
  Our second main result concerns the isometry groups of such distinguished
metrics. Among the completely solvable unimodular Lie groups (this includes
nilpotent groups), if the Lie group admits such a metric, we show that the
isometry group of this special metric is maximal among all isometry groups of
left-invariant metrics. We finish with a similar result for locally
left-invariant metrics on compact nilmanifolds.
