3X^4 + 3Y^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 3X^4 + 3Y^4 = 2Z^4 + W^4 has infinitely many integer solutions.

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

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

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

u^2 = k^4-10k^2+24k-17....................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 - x^2 + 35x + 37.
Rank is 1 and generator is [7 , 24].

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

Numerical example:

[X,Y,Z,W] = [25, 8, 19, 31], [17225, 4712, 18859, 10591]