49X^4 + 49Y^4 + 8Z^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 49X^4 + 49Y^4 + 8Z^4 = W^4 has infinitely many integer solutions.

49X^4 + 49Y^4 + 8Z^4 = W^4....................................................(1)
We use an identity 49(2(t+1))^4+49t^4+8(t^2+7t+8)^2 = (36+56t+29t^2)^2........(2)

So, we look for the integer solutions {Z^2 = t^2+7t+8, W^2 = 36+56t+29t^2}....(3)

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

u^2 =121k^4-912k^3+2640k^2-3264k+1444.........................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 - 2833x + 54168.
Rank is 2 and generator is [-12 , -294],[6521/16, -522291/64].

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

Numerical example:

[X,Y,Z,W] = [3689, 16892, 17818, 46819], [209239, 185468, 575318, 1007011]