9X^4 + 4Y^4 + 2Z^4 = 6W^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 + 4Y^4 + 2Z^4 = 6W^4 has infinitely many integer solutions.

9X^4 + 4Y^4 + 2Z^4 = 6W^4.....................................................(1)
We use an identity 9(2(t+1))^4+4t^4+2(t^2+6t+6)^2 = 6(5t^2+10t+6)^2...........(2)

So, we look for the integer solutions {Z^2 = t^2+6t+6, W^2 =5t^2+10t+6}.......(3)

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

u^2 = k^4+18k^2-80k+81........................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 -27x + 46.
Rank is 1 and generator is [-4 , -6].

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

Numerical example:

[X,Y,Z,W] = [72, 355, 239, 329],[22320, 64681, 96239, 79841]