2X^4 + 8Y^4 + Z^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 2X^4 + 8Y^4 + Z^4 = 9W^4 has infinitely many integer solutions.
This equation is related to 4X^4 + Y^4 + 2Z^4 = 18W^4.

2X^4 + 8Y^4 + Z^4 = 9W^4...........................................................(1)
We use an identity 2(6(t+1))^4+8(t)^4+(t^2+18t+18)^2=9(17t^2+34t+18)^2.............(2)

So, we look for the integer solutions {Z^2 = t^2+18t+18, W^2 =17t^2+34t+18}........(3)

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

u^2 =k^4+66k^2-1088k+4353..........................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 -1179x + 6858.
Rank is 1 and generator is [69 , 504].

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

Numerical example:

[X,Y,Z,W] = [30, 1, 19, 21],[11970, 449, 9379, 8581].