  Optimizing translationally invariant infinite-Projected Entangled Pair States
(iPEPS), we investigate the spin-2 Affleck-Kennedy-Lieb-Tasaki (AKLT) and
spin-1 Heisenberg models on the Kagome lattice as a function of magnetic field.
We found that the magnetization curves offer a wide variety of compressible and
incompressible phases. Incompressible nematic phases breaking the lattice $C_3$
rotation -- for which we propose simple qualitative pictures -- give rise to
magnetization plateaux at reduced magnetization $m_z=5/6$ and $m_z=1/3$ for
spin-2 and spin-1, respectively, in addition to the $m_z=0$ plateaux
characteristic of zero-field gapped spin liquids. Moving away from the plateaux
we observe a rich variety of compressible superfluid nematic -- named
"supernematic" -- phases breaking spontaneously both point group and spin-U(1)
symmetries, as well as a superfluid phase preserving lattice symmetries. We
also identify the nature -- continuous or first-order -- of the various phase
transitions. Possible connections to experimental spin-1 systems are discussed.
