The billiard motion inside an ellipsoid of ${\bf R}^{3}$ is completely integrable. If the ellipsoid is not of revolution, there are many orbits bi-asymptotic to its major axis. The set of bi-asymptotic orbits is described from a geometrical, dynamical, and topological point of view. It contains eight surfaces, called separatrices. The splitting of the separatrices under symmetric perturbations of the ellipsoid is studied using a symplectic discrete version of the Poincar\'e-Melnikov method, with a special emphasis in the following situations: close to the flat limit (when the minor axis of the ellipsoid is small enough), close to the oblate limit (when the ellipsoid is close to an ellipsoid of revolution around its minor axis) and close to the prolate limit (when the ellipsoid is close to an ellipsoid of revolution around its major axis). It is proved that any non-quadratic entire symmetric perturbation breaks the integrability and splits the separatrices, although (at least) sixteen symmetric homoclinic orbits persist. Close to the flat limit, these orbits become transverse under very general polynomial perturbations of the ellipsoid. Finally, a particular quartic symmetric perturbation is analyzed in great detail. Close to the flat and to the oblate limits, the sixteen symmetric homoclinic orbits are the unique primary homoclinic orbits. Close to the prolate limit, the number of primary homoclinic orbits undergoes infinitely many bifurcations. The first bifurcation curves are computed numerically.