  This brief review recalls some chapters in theory of sliding incommensurate
density waves which may have appeared after inspirations from studies of I.E
Dzyaloshinskii and collaborations with him. First we address the spin density
waves which rich order parameter allows for an unusual object of a complex
topological nature: a half-integer dislocation combined with a semi-vortex of
the staggered magnetization. It becomes energetically preferable with respect
to an ordinary dislocation due to the high Coulomb energy at low concentration
of carriers. Generation of these objects should form a sequence of pi-phase
slips in accordance with experimental doubling of phase-slips rate. Next, we
revise the commonly employed TDGL approach which is shown to suffer from a
violation of the charge conservation law resulting in nonphysical generation of
particles which is particularly pronounced for electronic vortices in the
course of their nucleation or motion. The suggested consistent theory exploits
the chiral transformations taking into account the principle contribution of
the fermionic chiral anomaly to the effective action. The derived equations
clarify partitions of charges,currents and rigidity among subsystems of the
condensate and normal carriers and the gluing electric field. Being
non-analytical with respect to the order parameter, contrarily the conventional
TDGL type, the resulting equations still allow for a numerical modeling of
transient processes related to space- and spatio-temporal vorticity in DWs.
