I have an ellipse in $x$-$y$ plane with eccentricity $e$, whose semi-major (along $x$-axis) and semi-minor (along $y$-axis) axes lengths are known to be $A$ and $B$. I rotate this ellipse about $y$-axis such that it makes an angle $\epsilon$ with the $x$-$y$ plane. Let's call this new ellipse $E_1$ (shown in red). Further, I rotate $E_1$ by $120^\circ$ anticlockwise about the $z$-axis, and I call this ellipse $E_2$ (shown in blue).
The problem is to find the point on $E_2$ that is at a distance of $L$ from the end-point of the semi-major axis of $E_1$.
The known quantities are: $e$ (eccentricity), the angle $\epsilon$, the coordinates $(x_0,y_0,z_0)$ of the end of semi-major axis of $E_1$, and the distance $L$. The ellipses are shown in the diagram.