Integer solutions to $\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=4$ A serious challenge:

Can someone find 3 positive whole numbers that solve this equation?
$$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=4$$

The numbers must be whole!
An answer has been given by Michael Stoll at this MO-question, linked at this MSE-duplicate, using the elliptic curve

$$E_n \colon y^2 = x \bigl(x^2 + (4n(n+3)-3)x + 32(n+3)\bigr)
                  =: x(x^2 + Ax + B),$$

where in our case $n=4$. Then the curve is known to have rank $1$, and thus there is a solution in positive integers of "truly enormous size", see the article "An Unusual Cubic Representation Problem" by Andrew Bremner. It is given by
$$
x = 437361267792869725786125260237139015281653755816161361862143‌​7993378423467772036;
$$

$$
 y = 368751317941299998271978115652254748254929799689719709962831‌37471637224634055579‌​; 
$$

$$
z = 154476802108746166441951315019919837485664325669565431700026‌63489825320203527799‌​9
$$ 
There are other solutions in positive integers, of course, but this is the smallest one.
$x=11$, $y=9$, $z=-5$

I brute forced it.