We investigate the asymptotic behaviour of the distribution of the number ξ(*B*) of the
solutions of a system of homogeneous random linear equations *Ax* = 0 (the *T* × *n* matrix *A* is composed of independent random variables *a*
_{i,j} uniformly distributed on a set of elements of a finite field *K*) which belong to some given set *B* of non-zero *n*-dimensional vectors over the field *K*. We consider the case where, under a concordant growth of the parameters *n*, *T* → ∞ and variations of the sets *B*
_{1}, . . . ,*B _{s}
* such that the mean values converge to finite limits, the limit distribution of the vector (ξ(

*B*

_{1}), ... , ξ(

*B*)) is an

_{s}*s*-dimensional compound Poisson distribution. We give sufficient conditions for this convergence and find parameters of the limit distribution. We consider in detail the special case where

*B*is the set of vectors which do not contain a certain element

_{k}*k*∈

*K*.

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