On the x-coordinates of Pell equations which are sums 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 $ such that the Pell equations $ x^2-dy^2 = \pm 1 $, $ X^2-dY^2=\pm 4 $ have at least two positive integer solutions $ (x,y) $ and $(x^{\prime}, y^{\prime})$, $ (X,Y) $ and $(X^{\prime}, Y^{\prime})$, respectively, such that each of $ x, ~x^{\prime}, ~X, ~X^{\prime} $ is a sum of two Padovan numbers.