Let $\pi$ be some set of primes, and let $G$ be the $\pi$-divisible $\pi$-torsion-free locally nilpotent group. For a system of equations over $G$ \begin{eqnarray*} f_1(x_1,\dots, x_n; a_1,\dots,a_m)=1,\\ ..............................\\ f_n(x_1,\dots, x_n; a_1,\dots,a_m)=1, \end{eqnarray*} let $\ell_{ij}$ be the sum of exponents of $x_j$ in the word $f_i$ (for all inclusions $x_j$ in $f_i$). If det $(\ell_{ij})$ is $\pi$-number, then this system has in $G$ the unique solution.