  We develop the approach to the problem of neutrino oscillations in a magnetic
field introduced in \cite{Popov:2019nkr} and extend it to the case of three
neutrino generations. The theoretical framework suitable for computation of the
Dirac neutrino spin, flavour and spin-flavour oscillations probabilities in a
magnetic field is given. It is shown that there is an entanglement between
neutrino flavour and spin oscillations and in the general case it is not
possible to consider these two types of neutrino oscillations separately. The
closed analytic expressions for the probabilities of oscillations are obtained
accounting for the normal and inverted hierarchies and the possible effect of
CP violation. In particular, it is shown that the probabilities of the
conversions without neutrino flavor change, i.e. $\nu_e^L \rightarrow \nu_e^L$
and $\nu_e^L \rightarrow \nu_e^R$, do not exhibit the dependence on the CP
phase, while the other neutrino conversions are affected by the CP phase. In
general, the neutrino oscillation probabilities exhibit quite a complicated
interplay of oscillations on the magnetic $\mu_{\nu} B$ and vacuum frequencies.
The obtained results are of interest in applications to neutrino oscillations
under the influence of extreme astrophysical environments, for example peculiar
to magnetars and supernovas, as well as in studying neutrino propagation in
interstellar magnetic fields.
