  Multiple-$Q$ magnetic states, such as a skyrmion crystal, become a source of
unusual transport phenomena and dynamics. Recent theoretical and experimental
studies clarify that such multiple-$Q$ states ubiquitously appear under
different crystal structures in metals and insulators. Toward a systematic
understanding of the formation of the multiple-$Q$ states in various crystal
systems, in this theoretical study, we present a low-energy effective spin
model with anisotropic exchange interactions in momentum space. We summarize
specific six symmetry rules for nonzero symmetric and antisymmetric anisotropic
exchange interactions in momentum space, which are regarded as an extension of
Moriya's rule. According to the rules, we construct the effective spin model
for tetragonal, hexagonal, and trigonal magnets with crystal- and
momentum-dependent anisotropic exchange interactions based on magnetic
representation analysis. Furthermore, we describe the origin of the effective
anisotropic exchange interactions in itinerant magnets by perturbatively
analyzing a multi-band periodic Anderson model with the spin-orbit coupling. We
apply the effective spin model to an itinerant magnet in a $P6/mmm$ crystal and
find various multiple-$Q$ states with a spin scalar chirality in the ground
state. Our results provide a foundation of constructing effective
phenomenological spin models for any crystal systems hosting the multiple-$Q$
states, which will stimulate further exploration of exotic multiple-$Q$ states
in materials with the spin-orbit coupling.
