18X^4 + 8Y^4 + Z^4 = 27W^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 18X^4 + 8Y^4 + Z^4 = 27W^4 has infinitely many integer solutions.
This equation is related to 36X^4 + Y^4 + 2Z^4 = 54W^4.

18X^4 + 8Y^4 + Z^4 = 27W^4................................................(1)
We use an identity 18(t+3)^4+8(t)^4+(t^2-27)^2 = 27(t^2+4t+9)^2...........(2)

So, we look for the integer solutions {Z^2 = t^2-27, W^2 =t^2+4t+9}.......(3)

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

u^2 = 69k^4-96k^3+90k^2-48k+21............................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 - 243x -1242.
Rank is 1 and generator is [21 , 54].

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

Numerical example:

[X,Y,Z,W] = [43, 91, 37, 69],[338707, 216163, 40931, 311549].