  We explore the field-temperature phase diagram of the XY pyrochlore
antiferromagnet Er$_2$Ti$_2$O$_7$, by means of magnetization and neutron
diffraction experiments. Depending on the field strength and direction relative
to the high symmetry cubic directions $[001], [1\bar{1}0]$ and $[111]$, the
refined field induced magnetic structures are derived from the zero field
$\psi_2$ and $\psi_3$ states of the $\Gamma_5$ irreducible representation which
describes the ground state of XY pyrochlore antiferromagnets. At low field,
domain selection effects are systematically at play. In addition, for $[001]$,
a phase transition is reported towards a $\psi_3$ structure at a characteristic
field $H_c^{001}=$ 43 mT. For $[1\bar{1}0]$ and $[111]$, the spins are
continuously tilted by the field from the $\psi_2$ state, and no phase
transition is found while domain selection gives rise to sharp anomalies in the
field dependence of the Bragg peaks intensity. For $[1\bar{1}0]$, these results
are confirmed by high resolution inelastic neutron scattering experiments,
which in addition allow us to determine the field dependence of the spin gap.
This study agrees qualitatively with the scenario proposed theoretically by
Maryasin {\it et al.} [Phys. Rev. B {\bf 93}, 100406(R) (2016)], yet the
strength of the field induced anisotropies is significantly different from
theory.
