Following problem though not a research problem
if $x,y,z,w$ are postive integers,and such $$xyzw=504(x^2+y^2+z^2+w^2)$$
such example $(x,y,z,w)=(21,63,84,84)$ hold,
Now My problem there exist distinct postive integer solution? or find this equation all solution?
The basic idea for Markoff-Hurwitz type equations $$ x_1^2+\cdots+x_n^2=Ax_1x_2\cdots x_n $$ with $A\in\mathbb Z$ is that if you have a solution $(x_1,\ldots,x_n)$ in integers, then by fixing $n-1$ of the variables to be the given values, you get a monic quadratic equation for the last variable, so since there is one integer solution to that quadratic, there are two. Similarly, if $A$ is a rational number, as in your case, any solution in rational solutions $(x_1,\ldots,x_n)$ leads to $n$ new solutions in rational numbers by "flipping" one of the coordinates. (Of course, this is a lie, since various of the $x_1$ might be the same and/or you might get a quadratic with a single root. But roughly speaking, you tend to get $n$ new solutions.)
You're looking at $x^2+y^2+z^2+w^2=\frac{1}{504}xyzw$. Taking the solution that you give, $(x,y,z,w)=(21,63,84,84)$, we get the flipped solutions $$ (861,63,84,84),\quad (21,231,84,84),\quad \left(21,63,\frac{273}{2},84\right). $$ So only the first two have integer coordinates. Further flips yield quite large solutions such as $$ (861,11991,84,84),\quad (3213,231,84,84). $$ And indeed, it's easy enough to see that if $(x,y,84,84)$ is a solution in integers, then flipping the $x$ or the $y$ gives integer solutions, since setting $z=w=84$ in your original equation yields $x^2+y^2-14xy+14112$, the flip equations for the $x$ and $y$ coordinates are monic.
Try $(x=28, y=42, z=84, w=98)$ and $(x=13, y=84, z=204, w=221)$.
EDIT: It was a coincidence. The following are some more solutions that have no $84$:
$$\matrix{ \left\{ 15,78,3570,6246 \right\} ,& \left\{ 21,60,453,900 \right\} ,& \left\{ 21,60,900,1797 \right\} ,\cr \left\{ 29,35,11670, 13128 \right\} ,& \left\{ 30,42,69,90 \right\} ,& \left\{ 30,42,90,156 \right\} ,\cr \left\{ 30,42,156,300 \right\} ,& \left\{ 30,42,300,594 \right\} , &\left\{ 30,42,594,1185 \right\}\cr} $$
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.
