On systems of polynomially solvable linear equations with $k$-valued variables

A class of polynomially solvable systems of $m$ linear equations of $n$ $k$-valued variables is described. The exact and asymptotic formulae for the cardinal number $\nu_k(n,m)$ of the class are presented. In particular, if $n,m\to\infty$ so that $m/n=(1-1/k)+\omega n^{-1/2}$, where $\omega\to+\infty$ almost all of such systems with columns in general position are polynomially solvable.