Your equation is a special case of the generalized Markov equation $$ x_1^2+\dots +x_n^2=xx_1\dots x_n $$ studied by Hurwitz in detail: Über eine Aufgabe der unbestimmten Analysis, Archiv. der Math 11 (1907), 185-196. The paper is available online here. (Markov studied the case $n=3$ and $x=3$, in which case the solutions correspond to the worst approximable real numbers.)

Hurwitz determined all solutions of the general equation. More precisely, he reduced the solutions to finitely many basic solutions satisfying (15)-(16) in the paper, and he described how to get all solutions from the basic ones.

Your equation is a special case of the generalized Markov equation $$ x_1^2+\dots +x_n^2=xx_1\dots x_n $$ studied by Hurwitz in detail: Über eine Aufgabe der unbestimmten Analysis, Archiv. der Math 11 (1907), 185-196. The paper is available online here. (Markov studied the case $n=3$ and $x=3$, in which case the solutions correspond to the worst approximable real numbers.)

Hurwitz determined all solutions of the general equation. More precisely, he reduced the solutions to finitely many basic solutions satisfying (15)-(16) in the paper, and he described how to get all solutions from the basic ones.

Added. Upon second reading, your equation would correspond to $x=\frac{1}{504}$, which is not an integer. Still, I regard Hurwitz's work to be relevant.