  We present the latest results of our search for planets with HARPS-N at the
3.6 m Telescopio Nazionale Galileo under the Tracking Advanced Planetary
Systems project: an in-depth study of the 15 most Li abundant giants from the
PennState - Toru\'n Planet Search sample. Our goals are first, to obtain radial
velocities of the most Li-rich giants we identified in our sample to search for
possible low-mass substellar companions, and second, to perform an extended
spectral analysis to define the evolutionary status of these stars. Methods.
This work is based on high-resolution spectra obtained with the Hobby-Eberly
Telescope and its High Resolution Spectro- graph, and with the HARPS-N
spectrograph at the Telescopio Nazionale Galileo. Two stars, HD 181368 and HD
188214 , were also observed with UVES at the VLT to determine beryllium
abundances. We report i) the discovery of two new planetary systems around the
Li-rich giant stars: HD 238914 and TYC 3318-01333- 1 (a binary system); ii)
reveal a binary Li-rich giant, HD 181368 ; iii) although our current phase
coverage is not complete, we suggest the presence of planetary mass companions
around TYC 3663-01966-1 and TYC 3105-00152-1 ; iv) we confirm the previous
result for BD+48 740 and present updated orbital parameters, and v) we find a
lack of a relation between the Li enhancement and the Be abundance for the
stars HD 181368 and HD 188214 , for which we acquired blue spectra. We found
seven stars with stellar or potential planetary companions among the 15 Li-rich
giant stars. The binary star frequency of the Li-rich giants in our sample
appears to be normal, but the planet frequency is twice that of the general
sample, which suggests a possible connection between hosting a companion and
enhanced Li abundance in giant stars. We also found most of the companions
orbits to be highly eccentric.
