By the circle method, an asymptotic formula is obtained for the number of solutions of the system of equations $$\sum\limits_{j = 1}^s {q_{jk} x_j^2 = n_k } \left( {k = 1, \ldots ,2} \right)$$ , where qjk are given positive integers and nk are increasing integers, in terms of integers X1,...,Xs. Conditions of nontriviality of this formula are investigated.