  The effects of competing magnetic interactions in stabilizing different spin
configurations are drawing a renewed attention in order to both unveil emerging
topological spin textures and to highlight microscopic mechanisms leading to
their stabilization. The possible key role of the two-site exchange anisotropy
in selecting specific helicity and vorticity of skyrmionic lattices has only
recently been proposed. In this work we explore the phase diagram of a
frustrated localized magnet on a two-dimensional centrosymmetric triangular
lattice, focusing on the interplay between the two-ion anisotropy (TIA) and the
single-ion anisotropy (SIA). The effects of an external magnetic field applied
perpendicularly to the magnetic layer are also investigated. By means of Monte
Carlo simulations, we find a profusion of different spin configurations, going
from trivial to high-order Q skyrmionic and meronic lattices. In closer detail,
we find that a dominant role is played by the two-ion over the single-ion
anisotropy in determining the planar spin texture, whereas the strength and
sign of SIA, together with the magnitude of the magnetic field, tune the
perpendicular spin components, mostly affecting the polarity (and, in turn, the
topology) of the spin-texture. Our analysis confirms the crucial role of
anisotropic symmetric exchange in systems with dominant short-range
interactions, at the same time predicting a rich variety of complex magnetic
textures that may arise from a fine tuning of competing anisotropic mechanisms.
