  We identified quasiparticle states at well-defined energies inside the
superconducting gap of the electron system at the LaAlO$_3$-SrTiO$_3$ interface
using tunneling spectroscopy. The states are found only in a number of samples
and depend upon the thermal-cycling history of the samples. The states consist
of a peak at zero energy and other peaks at finite energies, symmetrically
placed around zero energy. These peaks disappear, together with the
superconducting gap, with increasing temperature and magnetic field. We discuss
the likelihood of various physical mechanisms that are known to cause in-gap
states in superconductors and conclude that none of these mechanisms can easily
explain the results. The conceivable scenarios are the formation of Majorana
bound states, Andreev bound states, or the presence of an odd-frequency spin
triplet component in the superconducting order parameter.
