  We systematically investigate the effects of Kitaev interaction on magnetic
orders and anisotropy in both triangular and honeycomb lattices. Our study
highlights the critical role of the Kitaev interaction in modulating phase
boundaries and predicting new phases, e.g., zigzag phase in triangular lattice
and AABB phase in honeycomb lattice, which are absent with pure Heisenberg
interactions. Moreover, we reveal the special state-dependent anisotropy of
Kitaev interaction, and develop a general method that can determine the
presence of Kitaev interaction in different magnets. It is found that the
Kitaev interaction does not induce anisotropy in some magnetic orders such as
ferromagnetic order, while can cause different anisotropy in other magnetic
orders. Furthermore, we emphasize that the off-diagonal $\Gamma$ interaction
also contributes to anisotropy, competing with the Kitaev interaction to
reorient spin arrangements. Our work establishes a framework for comprehensive
understanding the impact of Kitaev interaction on ordered magnetism.
