Integral non-group-theoretical modular categories of dimension $p^2{q}^2$

We construct all integral non-group-theoretical modular categories of dimension $p^2{q}^2$, where $p$ and $q$ are distinct prime numbers, establishing that a necessary and sufficient condition for their existence is that $p\mid q+1$, and their rank is $p^2+\frac{q^2-1}{p}$.