9X^4 + 36Y^4 + 4Z^4 = 9W^4

Existence of solution for diophantine equation ax^4 + by^4 + cz^4 + dw^4 = 0 are known if abcd is square number.
So, we are curious about whether above equation has a solution or not if abcd is not square number.
In particular, when does this equation have infinitely many integer solutions?

Parametric solutions of x^4 + ay^4 = z^4 + bt^4 are given below.
x^4 + ay^4 = z^4 + bt^4

We show diophantine equation 9X^4 + 36Y^4 + 4Z^4 = 9W^4 has infinitely many integer solutions.

9X^4 + 36Y^4 + 4Z^4 = 9W^4...................................................(1)
We use an identity 9(t+1)^4+36(t)^4+4(3t^2+3t)^2 = 9(1+2t+3t^2)^2............(2)

So, we look for the integer solutions {Z^2 = 3t^2+3t, W^2 = 1+2t+3t^2}.......(3)

By parameterizing the second equation and substituting the result to first equation, then we obtain quartic equation below.

u^2 =2k^4+9..................................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 - 72x.
Rank is 1 and generator is [9 , -9].

Hence we can obtain infinitely many integer solutions for equation (1).

Numerical example:

[X,Y,Z,W] = [1, 12, 6, 17], [82369, 13872, 58548, 84673]