Parametric solutions $ax^4+by^4=cz^4$

Question

Fermat proved that $x^4+y^4=z^4$ has no non-trivial solutions. I am sure that the diophantine equation below does have integer solutions if $a=b=c\neq \pm 1$ $$ax^4+by^4=cz^4$$ Now can one tell me how I can get all the possible integer solutions?

Answer 1

Theorem about absence of non-trivial solutions of equation $$x^4 + y^4 = z^4$$ is a partial case of Fermat Last Theorem for exponent 4. It is a consequence of the single Fermat theorem with published author's prove.

The original question is too general for a clear answer.

Answer 2

$ax^4+by^4=cz^4$ has parametric solution given below:

$$(a,b,c)=[(x^2),(2y^2-3x^2),(x^2+2y^2)]$$

Above has condition: $(z^2=y^2-x^2)$

Numerical solution for $(x,y,z)=(5,13,12)$ is given below,

$25(5)^4+263(13)^4=363(12)^4$