  We have measured neutron diffraction patterns in a single crystal sample of
the pyrochlore compound Er$_2$Ti$_2$O$_7$ in the antiferromagnetic phase
(T=0.3\,K), as a function of the magnetic field, up to 6\,T, applied along the
[110] direction. We determine all the characteristics of the magnetic structure
throughout the quantum critical point at $H_c$=2\,T. As a main result, all Er
moments align along the field at $H_c$ and their values reach a minimum. Using
a four-sublattice self-consistent calculation, we show that the evolution of
the magnetic structure and the value of the critical field are rather well
reproduced using the same anisotropic exchange tensor as that accounting for
the local paramagnetic susceptibility. In contrast, an isotropic exchange
tensor does not match the moment variations through the critical point. The
model also accounts semi-quantitatively for other experimental data previously
measured, such as the field dependence of the heat capacity, energy of the
dispersionless inelastic modes and transition temperature.
