  The chiral kinetic theory is derived from exact spinor mean field equations
without symmetry-breaking terms for large classes of SU(2) systems with
spin-orbit coupling. The influence of the Wigner function's off-diagonal
elements is worked out. The decoupling of the diagonal elements renormalizes
the drift according to Berry connection which is found as an expression of the
meanfield, spin-orbit coupling and magnetic field. As special limit, Weyl
systems are considered. The anomalous term $\sim\V E\V B$ in the balance of the
chiral density appears consequently by an underlying conserving theory. The
experimental observations of this term and the anomalous magneto-transport in
solid-sate physics usually described by chiral kinetic theory are therefore not
a unique signal for mixed axial-gravitational or triangle anomaly and no signal
for the breaking of Lorentz-invariance. The source of the anomalous term is by
two thirds the divergence of Berry curvature at zero momentum which can be seen
as Dirac monopole and by one third the Dirac sea at infinite momentum. During
the derivation of the chiral kinetic theory this source by the Dirac sea is
transferred exclusively to the Dirac monopole due to the projection of the
spinor Wigner functions to the chiral basis. The dynamical result is shown to
suppress the anomalous term by two thirds.
