7X^4 + 7Y^4 + 2Z^4 = 9W^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 + 2Z^4 = 9W^4 has infinitely many integer solutions.

7X^4 + 7Y^4 + 2Z^4 = 9W^4.....................................................(1)
We use an identity 7(3(t+1))^4+7(t)^4+2(t^2+9t+9)^2=9(8t^2+16t+9)^2...........(2)

So, we look for the integer solutions {Z^2 = t^2+9t+9, W^2 =8t^2+16t+9}.......(3)

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

u^2 =k^4+30k^2-224k+393.......................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 -117x + 324.
Rank is 1 and generator is [0 , -18].

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

Numerical example:

[X,Y,Z,W] = [264, 95, 59, 249],[17291235, 11635272, 32216937, 23843067].