Solve $ab(a^2+b^2)=2c^4$ over $\mathbb Z$

Question

Find all integer solutions to $ab\left(a^2+b^2\right)=2c^4$.

I have found four classes of solutions: $(n,n,n)$, $(n,n,-n)$, $(n,0,0)$ and $(0,n,0)$, for $n\in\mathbb Z$. I think these are all, but I'm not sure.

It suffices to look for positive integer solutions. Let $g=\gcd(a,b)$. Since $g^4$ divides the LHS, it follows that $g$ must divide $c$. So we can divide out by $g$ and look for solutions where $a,b$ are coprime. In this case, $\gcd(ab,a^2+b^2)=1$.

I am stuck from here though.

Answer 1

Multiply by $8$ and rearrange to get $$(a+b)^4=(2c)^4+(a-b)^4.$$ From here, I have discovered a truly wonderful solution, which math.SE is too small to contain.