On the $x$--coordinates of Pell equations that are products of two Padovan numbers

Let $ \{P_{n}\}_{n\geq 0} $ be the sequence of Padovan numbers defined by $ P_0=0 $, $ P_1 = P_2=1$, and $ P_{n+3}= P_{n+1} +P_n$ for all $ n\geq 0 $. In this paper, we find all positive square-free integers $ d \ge 2$ such that the Pell equations $ x^2-dy^2 = \ell$, where $ \ell\in\{\pm 1, \pm 4\} $, have at least two positive integer solutions $ (x,y) $ and $(x^{\prime}, y^{\prime})$ such that each of $ x$ and $x^{\prime}$ is a product of two Padovan numbers