  Positive semidefiniteness, recursiveness, and the variety condition of a
moment matrix are necessary and sufficient conditions to solve the quadratic
and quartic moment problems. Also, positive semidefiniteness, combined with
another necessary condition, consistency, is a sufficient condition in the case
of extremal moment problems, i.e., when the rank of the moment matrix (denoted
by r) and the cardinality of the associated algebraic variety (denoted by v)
are equal. However, these conditions are not sufficient for non-extremal sextic
or higher-order truncated moment problems.
  In this paper we settle three key instances of the non-extremal (i.e., r<v)
sextic moment problem, as follows: when r=7, positive semidefiniteness,
consistency and the variety condition guarantee the existence of a 7-atomic
representing measure; when r=8 we construct two determining algorithms,
corresponding to the cases v=9 and $v=+ \infty$. \ To accomplish this, we
generalize the rank-reduction technique developed in previous work, where we
solved the nonsingular quartic moment problem and found an explicit way to
build a representing measure.
