  Here, we find all positive integer solutions of the Diophantine equation in
the title, where $(\mathcal{U}_n)_{n\geqslant 0}$ is the generalized Lucas
sequence $\mathcal{U}_0=0, \ \mathcal{U}_1=1$ and $\mathcal{U}_{n+1}=r
\mathcal{U}_n +s \mathcal{U}_{n-1}$ with $r$ and $s$ integers such that $\Delta
= r^2 +4 s >0$.
