On the equation $x_1^2 + x_2^2 + x_3^2 + x_4^2 = N$ with variables such that $x_1 x_2 x_3 x_4 + 1$ is an almost-prime

Abstract: We consider Lagrange's equation $x_1^2 + x_2^2 + x_3^2 + x_4^2 = N$, where $N$ is a sufficiently large and odd integer, and prove that it has a solution in natural numbers $x_1, \dots, x_4 $ such that $x_1 x_2 x_3 x_4 + 1$ has no more than 48 prime factors.