33X^4 + 33Y^4 = 2Z^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 33X^4 + 33Y^4 = 2Z^4 + W^4 has infinitely many integer solutions.

33X^4 + 33Y^4 = 2Z^4 + W^4...................................................(1)
We use an identity 33(t+1)^4+33(t)^4-2(t^2+9t+4)^2 = (1-6t-8t^2)^2...........(2)

So, we look for the integer solutions {Z^2 = t^2+9t+4, W^2 =1-6t-8t^2}.......(3)

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

u^2 = k^4+24k^3-184k^2+552k-593..............................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 - x^2 + 4335x + 21825.
Rank is 2 and generator is [15 , 300],[47 , 572].

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

Numerical example:

[X,Y,Z,W] = [37, 20, 56, 83], [1445, 84, 2864, 1747],