  The influence of an external electric field on the spin dynamics of an
electrically neutral Fermi liquid is considered, the mechanism of such an
influence being the relativistic spin-orbital interaction. As a result,
Leggett's equations for the spin dynamics of weakly polarized Fermi liquids are
generalized to the case of non-zero external electric field. In addition, we
obtained the transverse spin dynamics equation for strongly spin-polarized
liquids in an electric field at zero temperature. In both situations covariant
derivatives depending on the electric field are shown to be substituted for
spatial gradients in line with the SU(2) gauge invariance of the microscopic
Hamiltonian. The new equations are applied to the study of spin flow along a
channel, where an electric field is found to bring about an additional phase
shift of the order of magnitude of the phase shift in superfluid $^3$He-$B$ but
growing with time.
