  Any group that has a subnormal series, in which all factors are abelian and
all except the last one are $p'$-torsion-free, can be embedded into a group
with a subnormal series of the same length, with the same properties and such
that any $p$-nonsingular system of equations over this group is solvable in
this group itself. This helps us to prove that the minimal order of a
metabelian group, over which there is a unimodular equation that is unsolvable
in metabelian groups, is 42.
