Consider an overdetermined system of nonlinear equations of the following form: $\pi_m(x)+g_m(y_1,\dots,y_N)=\pi_m(\tilde x)+g_m(\tilde y_1,\dots,\tilde y_N)$, where $m=1,\dots,M$ and $x,\tilde x, y_n, \tilde y_n$ are real numbers from $R$. The system is potentially overdetermined in the sense that $M>2N+2$, i.e., number of equations exceeds the number of unknowns. Functions $\pi_m(\cdot)$ and $g_m(\cdot)$ are non-negative.
I am trying to understand conditions on functions $\pi_m(\cdot)$, $g_m(\cdot)$ and dimensions $M$ and $N$ such that the set of all possible pairs $(x,\tilde x)\in R^2$, for which the system has solutions, has zero measure in $R^2$. Because the system is overdetermined, intuitively, it seems that this measure should be zero under some rather general conditions. Also, more generally, is there a theorem that establishes the measure of a set of solutions of overdetermined systems of equations?
I tried various ideas based on Sard's theorem but did not get anywhere... Would be most grateful for any help with this problem.