Is it possible to find intersection points of two ellipses analytically?

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We have two ellipses: first has semi-major and semi-minor axes parallel to coordinate axes with center at the coordinates origin, the second has random rotation and random center position. Is it possible to find intersection points (if exist) without numerical integration?

I tried to solve the system of two equations in Cartesian coordinates but the second equation has a sin and cos functions and I don't know how to solve it.