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

9X^4 + 9Y^4 = 2Z^4 + W^4.......................................................(1)

We use an identity 9(t+1)^4 + 9t^4 = 2(t^2-3t-2)^2 + (1+6t+4t^2)^2.............(2)

So, we look for the integer solutions {Z^2 = t^2-3t-2, W^2 = 4t^2+6t+1}........(3)

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

u^2 = -2k^4+6k^3+2k^2-48k+76...................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3-x^2 + 319x -975.
Rank is 1 and generator is [8 , 45].

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

Numerical example:

[X,Y,Z,W] = [41, 153, 223, 58], [854713, 1579017, 2207969, 1906742].