TITLE: $x^2+y^2+z^2=5(xy+yz+zx)$ -- Is this all solutions?
QUESTION [9 upvotes]: Problem. Find all integers $x$, $y$, and $z$ that satisfy $$x^2+y^2+z^2=5(xy+yz+zx)\,.$$

Does the following parametrization give all solutions?:
$$x=m^2+mn-5n^2\,,$$
$$y=-5m^2+9mn-3n^2\,,$$
$$z=-3m^2-3mn+n^2\,,$$
where $m,n$ are arbitrary integers.
Also, all permutations of this, and all permutations with each one multiplied by $-1$, e.g.
$$x=-m^2-mn+5n^2\,,$$
$$y=5m^2-9mn+3n^2\,,$$
$$z=3m^2+3mn-n^2\,,$$
etc.

REPLY [2 votes]: As enquired in the question at the top, there is also another parametric solution which is given below,
$x=(5p^2+12pq+3q^2)$
$y=(3p^2+24pq+41q^2)$
$z=(-p^2-8pq+5q^2)$