p$-Nonsingular systems of equations over solvable groups

Any group that has a subnormal series all factors in which are abelian and all factors 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. Using this we prove that the minimal order of a metabelian group over which there exists a unimodular equation that is unsolvable in metabelian groups is $42$. Bibliography: 14 titles.