diophantine equation with squares over 3 variables I am trying to find solutions for this diophantine equation

$$x^2+y^2+x^2y^2=4z^2$$

I am looking for advice on a procedure to find all positive integer solutions for this equations.
by considering x,y are both even and let $x^2=s ,y^2=r$

then $sr+s+r=(2z)^2$ this equation is equivalent to:

$(2s+r+1)^2=(r-1)^2+(2s)^2+(4z)^2$  you can check that
and the positive solutions for the last equation are given by the dimensions and the length of the diagonal of a rectangular box which is a related problem to Pythagorean Triple.

so $s=a , b=a+1 , r=(4z^2-a)/b$

&amp; b is a divisor of $a^2+4z^2  ,b&lt;√(a^2+4z^2 )   ,   1&lt;2z^2$

one can solve this equation over the positive integers if he just know the value of  $a$
by considering x,y are both even and let $x^2=s ,y^2=r$

then $sr+s+r=(2z)^2$ this equation is equivalent to:

$(2s+r+1)^2=(r-1)^2+(2s)^2+(4z)^2$  you can check that
and the positive solutions for the last equation are given by the dimensions and the length of the diagonal of a rectangular box which is a related problem to Pythagorean Triple.

so $s=a , b=a+1 , r=(4z^2-a)/b$

&amp; b is a divisor of $a^2+4z^2  ,b&lt;√(a^2+4z^2 )   ,   1&lt;2z^2$

one can solve this equation over the positive integers if he just know the value of  $a$