  We investigate the class of regular-ordered word equations. In such
equations, each variable occurs at most once in each side and the order of the
variables occurring in both sides is the preserved (the variables can be,
however, separated by potentially distinct constant factors). Surprisingly, we
obtain that solving such simple equations, even when the sides contain exactly
the same variables, is NP-hard. By considerations regarding the combinatorial
structure of the minimal solutions of the more general quadratic equations we
obtain that the satisfiability problem for regular-ordered equations is in NP.
Finally, we also show that a related class of simple word equations, that
generalises one-variable equations, is in P.
