It is known that the problem of determining consistency of a
finite system of equations in a free group or a free monoid is
decidable, but the corresponding problem for systems of equations
in a free inverse monoid of rank at least two is undecidable. Any
solution to a system of equations in a free inverse monoid
induces a solution to the corresponding system of equations in the
associated free group in an obvious way, but solutions to systems
of equations in free groups do not necessarily lift to solutions
in free inverse monoids. In this talk we show that the problem of
determining whether a solution to a finite system of equations in
a free group can be extended to a solution of the corresponding system in the associated free inverse monoid is decidable. We are
able to use this, combined with some results of Tim Deis, to solve the
consistency problem for certain
classes of single variable equations in free inverse monoids.

Sponsored in part by the FCT approved
projects POCTI 32817/99 and POCTI/MAT/37670/2001 in participation with
the European Community Fund FEDER and by FCT through *Centro de Matemática
da Universidade do Porto.* Also sponsored in part by FCT, the
*Faculdade de Ciências da Universidade do Porto, Programa Operacional Ciência, Tecnologia, Inovação do Quadro Comunitário de Apoio III*, and by *Caixa Geral de Depósitos*.