27X^4 + 12Y^4 = 4Z^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 27X^4 + 12Y^4 = 4Z^4 + 3W^4 has infinitely many integer solutions.
This equation is related to X^4 + 36Y^4 = 12Z^4 + 9W^4.

27X^4 + 12Y^4 = 4Z^4 + 3W^4....................................................(1)
We use an identity 27(t+1)^4+12(t)^4-4( 3t^2+3t)^2 = 3(t^2+6t+3)^2.............(2)

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

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

u^2 =-2k^4+27..................................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 - 216x.
Rank is 1 and generator is [25 , -145].

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

Numerical example:

[X,Y,Z,W] = [3251043, 3455881, 5805657, 2796715], [7704517978502370723, 381920211042841, 93954954084385377, 13344616575663486475]