I have a system of $n$ congruences.

the generic $m$_th congruence of the system ($m = 1,\dots,n$) is in the form:

$(p_n\#)^2(\displaystyle\sum_{\substack{i=1\\i\neq m}}^n{\frac{x_iy_m}{p_ip_m}}+\sum_{\substack{j=1\\j\neq m}}^n{\frac{x_my_i}{p_jp_m}}+\frac{x_my_m}{p_m^2}-n_2\frac{x_m}{p_m}-n_1\frac{y_m}{p_m})\equiv b_m \:\:\: (mod \:\: p_m^2)$

where:

$p_n\#=\displaystyle\prod_{\substack{i=1}}^np_i$ and $p_i$ is the $i$_th prime number.

$x_i$ and $y_i$ are the unknows while $n_1$ and $n_2$ are known.

for every couple of unknowns $(x_m,y_m)$ $(m=1,\dots,n)$ i know $p_m-1$ couple of values where the solution lies, therefore, in order to avoid nonlinearity, for every congruence of the system i can build a set of $p_m-1$ linear congruences in which the values of $(x_m,y_m)$ are fixed. The problem now is to choose the correct combination of linear congruences (one for each set) in order to build a consistent system. Every congruence of the complete system should be $0\equiv0\:\:(mod\:p_m^2)$ because every time i choose a congruence from the $m$_th set, i choose implicitly the values of the couple of unknowns $(x_m,y_m)$.

I know that there is at least a solution (and i am interested in finding a single solution) but i'm afraid of not being able to do an exaustive search over the linear congruences because the size of the problem is too big (30+ nonlinear congruences resulting in a 30+ set of linear congruences).

Any help would be appreciated. Thanks.