  We study the angular dependence in polarized light of the optical absorption
for the Er$^{3+}$ transition $^{4}I_{15/2} \rightarrow $$^{4}I_{13/2}$ in
Y$_2$SiO$_5$, revealing thus the associated anisotropy and the orientation of
the related absorption principal directions in the dielectric plan
perpendicular to the monoclinic axis b. The measurements are performed at low
temperature. This allows us to isolate the lowest crystal field levels in the
ground and excited states. We spectrally resolve and independently characterize
the two yttrium substitution sites in the Y$_2$SiO$_5$ matrix. The absorption
tensor components cannot be unambiguously determined yet while only considering
the investigated dielectric plane. Still, measurements remarkably demonstrate
that this transition of interest well resolved at low temperature is not only a
magnetic-dipole allowed transition but indeed a hybrid electric-magnetic
transition.
