On the $x-$coordinates of Pell equations which are sums of two Padovan numbers

Let $\{P_n{\}}_{n\ge 0}$ 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 $n\ge 0$. In this paper, we find all positive square-free integers $d$ such that the Pell equations $x^2-d{y}^2=\pm 1$, $X^2-d{Y}^2=\pm 4$ have at least two positive integer solutions $(x,y)$ and $(x^{\mathrm{\prime }},y^{\mathrm{\prime }})$, $(X,Y)$ and $(X^{\mathrm{\prime }},Y^{\mathrm{\prime }})$, respectively, such that each of $ x, ~x^{\prime}, ~X, ~X^{\prime} $ is a sum of two Padovan numbers.