  We introduce the notion of idempotent variables for studying equations in
inverse monoids.
  It is proved that it is decidable in singly exponential time (DEXPTIME)
whether a system of equations in idempotent variables over a free inverse
monoid has a solution. The result is proved by a direct reduction to solve
language equations with one-sided concatenation and a known complexity result
by Baader and Narendran: Unification of concept terms in description logics,
2001. We also show that the problem becomes DEXPTIME hard , as soon as the
quotient group of the free inverse monoid has rank at least two.
  Decidability for systems of typed equations over a free inverse monoid with
one irreducible variable and at least one unbalanced equation is proved with
the same complexity for the upper bound.
  Our results improve known complexity bounds by Deis, Meakin, and Senizergues:
Equations in free inverse monoids, 2007.
  Our results also apply to larger families of equations where no decidability
has been previously known.
