Intersections of binary quadratic forms in smooth numbers

The number of solutions to $a^{2}+b^{2}=c^{2}+d^{2}\le x$ in integers is a well-known result, while if one restricts all the variables to primes Erd\H{o}s[4] showed that only the diagonal solutions, namely, the ones with $\{a,\,b\}=\{c,\,d\}$ contribute to the main term, hence there is a paucity of the off-diagonal solutions. A natural number $n$ is $y$-smooth if the greatest prime factor of $n$ does not exceed $y$. In this paper, we restrict some of the variables to the smooth number.