X^4 + 36Y^4 + 27Z^4 = 4W^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 X^4 + 36Y^4 + 27Z^4 = 4W^4 has infinitely many integer solutions.

X^4 + 36Y^4 + 27Z^4 = 4W^4....................................................(1)
We use an identity (3(t+1))^4+36t^4+27(t^2+2t+3)^2 = 4(6t^2+9t+9)^2...........(2)

So, we look for the integer solutions {Z^2 = t^2+2t+3, W^2 = 6t^2+9t+9}.......(3)

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

u^2 =306k^4-1296k^3+2340k^2-1728k+474.........................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 - 648x + 1728.
Rank is 1 and generator is [-8 , -80].

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

Numerical example:

[X,Y,Z,W] = [51693, 41613, 38547, 81294], [4685241771213, 6440080069293, 7073564628147, 13426812048894]