9X^4 + Y^4 + 2Z^4 = 3W^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 + Y^4 + 2Z^4 = 3W^4 has infinitely many integer solutions.
This equation is related to X^4 + 9Y^4 + 18Z^4 = 27W^4.

9X^4 + Y^4 + 2Z^4 = 3W^4..........................................................(1)

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

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

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

u^2 = 59k^4-280k^3+534k^2-448k+137................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 - 24x + 32.
Rank is 1 and generator is [-4 , -8].

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

Numerical example:

[X,Y,Z,W] = [40, 47, 37, 57], [84360, 121777, 73213, 125033].