  A conservative class of constraint satisfaction problems CSPs is a class for
which membership is preserved under arbitrary domain reductions. Many
well-known tractable classes of CSPs are conservative. It is well known that
lexleader constraints may significantly reduce the number of solutions by
excluding symmetric solutions of CSPs. We show that adding certain lexleader
constraints to any instance of any conservative class of CSPs still allows us
to find all solutions with a time which is polynomial between successive
solutions. The time is polynomial in the total size of the instance and the
additional lexleader constraints. It is well known that for complete symmetry
breaking one may need an exponential number of lexleader constraints. However,
in practice, the number of additional lexleader constraints is typically
polynomial number in the size of the instance. For polynomially many lexleader
constraints, we may in general not have complete symmetry breaking but
polynomially many lexleader constraints may provide practically useful symmetry
breaking -- and they sometimes exclude super-exponentially many solutions. We
prove that for any instance from a conservative class, the time between finding
successive solutions of the instance with polynomially many additional
lexleader constraints is polynomial even in the size of the instance without
lexleaderconstraints.
