MHB Solve triple square Diophantine equation

[individ]

Once you know how to solve it, then explain how to solve Diophantine equation:

$$X^2+Y^2=aZ^2$$

$$a$$ - integer. Write the equation when it has a solution.

 

[individ]

Thank you!
When you need an answer I will give a link to it.

 

[individ]

For example for such equation:

$$y^2+ax^2=z^2$$

The solutions have the form:

$$y=p^2-as^2$$

$$x=2ps$$

$$z=p^2+as^2$$

For example for such equation:

$$y^2+ax^2=az^2$$

The solutions have the form:

$$y=2aps$$

$$x=ap^2-s^2$$

$$z=ap^2+s^2$$

$$p,s$$ - integers.
But for such equations, of which I said, to write a simple formula is impossible. I wonder why?

 

[individ]

You can write and easy solution and similar equation:

number theory - Find all integers satisfying $m^2=n_1^2+n_1n_2+n_2^2$ - Mathematics Stack Exchange

There are many formulas to see. The questions will be answered here.