We summarize a proof for the existence and uniqueness of the loxodrome joining two distinct points $p_0$ and $p_1$ on an open half of an ellipsoid of revolution. We also compute the unique angle $\alpha \in [0,2\pi)$ which the loxodrome makes with the meridians intersecting the loxodrome.
Abed Elhashash. "Existence, Uniqueness and Angle Computation for the Loxodrome on an Ellipsoid of Revolution." J. Geom. Symmetry Phys. 13 75 - 88, 2009. https://doi.org/10.7546/jgsp-13-2009-75-88