7X^4 + 7Y^4 + Z^4 = 8W^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 7X^4 + 7Y^4 + Z^4 = 8W^4 has infinitely many integer solutions.
This equation is related to 14X^4 + 14Y^4 + 2Z^4 = W^4.

7X^4 + 7Y^4 + Z^4 = 8W^4..........................................................(1)
We use an identity 7(t+4)^4+7(2t)^4-8(4(t^2+t+4))^2 = -(3t^2+24t-16)^2............(2)

So, we look for the integer solutions {Z^2 = 3t^2+24t-16, W^2 = 4t^2+4t+16}.......(3)

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

u^2 =k^4+12k^3-26k^2-36k+37.......................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 + x^2 - 50x + 48.
Rank is 1 and generator is [-2 , -12].

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

Numerical example:

[X,Y,Z,W] = [48, 5, 47, 37],[11395, 83472, 87023, 104413]