An Elliptic curve for solving $W^4+X^4=Y^4+Z^4$

It is well known that a parameterization for the equation $$W^2+X^2=Y^2+Z^2$$ is given by:

$$(W,X,Y,Z)=(a+b,ab-1,ab+1,a-b).$$

I have found a similar type of parameterization for the fourth-power equation: $$W^4+X^4=Y^4+Z^4$$ where $W = d+e$, $X = cde-1$, $Y = d-e$, and $Z = cde+1$, provided that the elliptic curve $$y^2 = x^3 - 27(c^6+14c^3+1)x - 54(c^9-33c^6-33c^3-1)$$ possesses a rational point $(x, y)$ given by: $$x = \frac{18 + 18d - 3(1 + c^3 + 6c^3d)e^2}{e^2},\qquad y = \frac{-6x + 36c^3 + 36}{e}$$

I am unable to find an initial rational point using only pen-and-paper methods. While I understand this is typically computed using a computer algebra system (CAS), I currently do not have access to one.

Could anyone help me find the initial rational point, as well as any additional rational points?

1. Can we find solutions where the sum of two odd fourth powers is equal to another sum of two odd fourth powers?
2. Can the sum of two odd fourth powers be expressed as a sum of two even fourth powers?