  We present an experimental study of the quantum spin ice candidate pyrochlore
coumpound \przr\ by means of magnetization measurements, specific heat and
neutron scattering up to 12 T and down to 60 mK. When the field is applied
along the $[111]$ and $[1\bar{1}0]$ directions, ${\bf k}=0$ field induced
structures settle in. We find that the ordered moment rises slowly, even at
very low temperature, in agreement with macroscopic magnetization.
Interestingly, for $H \parallel [1\bar{1}0]$, the ordered moment appears on the
so called $\alpha$ chains only. The spin excitation spectrum is essentially
{\it inelastic} and consists in a broad flat mode centered at about 0.4 meV
with a magnetic structure factor which resembles the spin ice pattern. For $H
\parallel [1\bar{1}0]$ (at least up to 2.5 T), we find that a well defined mode
forms from this broad response, whose energy increases with $H$, in the same
way as the temperature of the specific heat anomaly. We finally discuss these
results in the light of mean field calculations and propose a new
interpretation where quadrupolar interactions play a major role, overcoming the
magnetic exchange. In this picture, the spin ice pattern appears shifted up to
finite energy because of those new interactions. We then propose a range of
acceptable parameters for \przr\, that allow to reproduce several experimental
features observed under field. With these parameters, the actual ground state
of this material would be an antiferroquadrupolar liquid with spin-ice like
excitations.
