  The atmospheric characterisation of hot and warm Neptune-size exoplanets is
challenging due to their small radius and atmospheric scale height. The
warm-Neptune HAT-P-11b is a remarkable target for such characterisation due to
the large brightness of its host star (V=9.46 mag; H=7.13 mag). The aims of
this work are to review the main physical and architectural properties of the
HAT-P-11 planetary system, and to probe the presence of 8 molecular species in
the atmosphere of HAT-P-11b at high spectral resolution in the near-infrared.
The planetary system was reviewed by analysing transits and occultations of
HAT-P-11b from the Kepler data set as well as HIRES at Keck archival
radial-velocity (RV) data. We modelled the latter with Gaussian-process
regression and a combined quasi-periodic and squared-exponential kernel to
account for stellar variations on both (short-term) rotation and (long-term)
activity-cycle timescales. In order to probe the atmospheric composition of
HAT-P-11b, we observed 4 transits of this target with GIANO-B at TNG. We find
that the long-period ($P\sim9.3$ years) RV signal previously attributed to
planet HAT-P-11c is more likely due to the stellar magnetic activity cycle.
Nonetheless, the Hipparcos-Gaia difference in the proper-motion anomaly
suggests that an outer-bound companion might still exist. For HAT-P-11b, we
measure a radius $R_{\rm p}=0.4466\pm0.0059\,R_{\rm J}$, a mass $M_{\rm
p}=0.0787\pm0.0048\,M_{\rm J}$, and an eccentricity
$e=0.2577^{+0.0033}_{-0.0025}$, in accordance with values in the literature.
Probing its atmosphere, we detect $NH_3$ (S/N$=5.3$, significance$=5.0\sigma$)
and confirm the presence of $H_2O$ (S/N$=5.1$, significance$=3.4\sigma$). We
also tentatively detect the signal of $CO_2$ (S/N$=3.0$,
significance$=3.2\sigma$) and $CH_4$ (S/N$=4.8$, significance$=2.6\sigma$),
whose presence need to be confirmed by further observations.
