  We give NSPACE(n log n) algorithms solving the following decision problems.
Satisfiability: Is the given equation over a free partially commutative monoid
with involution (resp. a free partially commutative group) solvable?
Finiteness: Are there only finitely many solutions of such an equation? PSPACE
algorithms with worse complexities for the first problem are known, but so far,
a PSPACE algorithm for the second problem was out of reach. Our results are
much stronger: Given such an equation, its solutions form an EDT0L language
effectively representable in NSPACE(n log n). In particular, we give an
effective description of the set of all solutions for equations with
constraints in free partially commutative monoids and groups.
