  In this article, we study the combinatorics of congruence subgroups of the
modular group. More precisely, we consider the notion of minimal monomial
solutions. These are the solutions of a matrix equation (also appearing in the
study of Coxeter friezes), modulo an integer $N$, whose components are
identical and minimal for this property. The objective here is to characterize
the solutions of this type verifying a property of irreducibility modulo a
prime power.
