Solving gauge anomaly equations in the Standard Model using the Method of Chords

In a recent paper, Allanach et al introduced geometric method to solve the anomaly cancellation equations for $U(1)$ gauge theory with an arbitrary number of charges - Method Chords known in Diophantine analysis. We extend their result non-Abelian groups, and show that this can be used find general solution Standard Model group on curved background. Given $K$ $(\textbf{2}, \textbf{3})$, $L$ \textbf{1})$, $M$ $(\textbf{1}, $N$ \textbf{1})$ representations $SU(2)\times SU(3)$, reduce homogeneous cubic equation $K+L+M+N-4$ variables.