25X^4 + Y^4 + 10Z^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 25X^4 + Y^4 + 10Z^4 = W^4 has infinitely many integer solutions.

25X^4 + Y^4 + 10Z^4 = W^4....................................................(1)
We use an identity 25(t+1)^4+(2t)^4+10(2t^2+2t)^2 = (5+10t+9t^2)^2...........(2)

So, we look for the integer solutions {Z^2 =2t^2+2t, W^2 = 5+10t+9t^2}.......(3)

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

u^2 = 4k^4+20................................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 - 20x.
Rank is 1 and generator is [-4 , -4].

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

Numerical example:

[X,Y,Z,W] = [4, 1, 2, 9], [1296, 6241, 2844, 6881]