Non-negative integer linear congruences

We consider the problem of describing all non-negative integer solutions to a linear congruence in many variables. This question may be reduced to solving the congruence x1 + 2x2 + 3x3 +···+(n−1)xn−1 ≡ 0 (mod n) where xi ∈ N ={0, 1, 2,...}. We consider the monoid of solutions of this equation and prove equivalent two conjectures of Elashvili concerning the structure of these solutions. This yields a simple algorithm for generating most (conjecturally all) of the high degree indecomposable solutions of the equation