  We consider nuclear spin dynamics in a two-electron double dot system near
the intersection of the electron spin singlet $S$ and the lower energy
component $T_{+}$ of the spin triplet. The electron spin interacts with nuclear
spins and is influenced by the spin-orbit coupling. Our approach is based on a
quantum description of the electron spin in combination with the coherent
semiclassical dynamics of nuclear spins. We consider single and double
Landau-Zener passages across the $S$-$T_{+}$ anticrossings. For linear sweeps,
the electron dynamics is expressed in terms of parabolic cylinder functions.
The dynamical nuclear polarization is described by two complex conjugate
functions $\Lambda ^{\pm}$ related to the integrals of the products of the
singlet and triplet amplitudes ${\tilde{c}}_{S}^{\ast}{\tilde{c}}_{T_{+}}$
along the sweep. The real part $P$ of $\Lambda ^{\pm}$ is related to the
$S$-$T_{+}$ spin-transition probability, accumulates in the vicinity of the
anticrossing, and for long linear passages coincides with the Landau-Zener
probability $P_{LZ}=1-e^{-2\pi \gamma}$, where $\gamma $ is the Landau-Zener
parameter. The imaginary part $Q$ of $\Lambda^{+}$ is specific for the nuclear
spin dynamics, accumulates during the whole sweep, and for $\gamma \gtrsim 1$
is typically an order of magnitude larger than $P$. $Q$ has a profound effect
on the nuclear spin dynamics, by (i) causing intensive shake-up processes among
the nuclear spins and (ii) producing a high nuclear spin generation rate when
the hyperfine and spin-orbit interactions are comparable in magnitude. We find
analytical expressions for the back-action of the nuclear reservoir represented
via the change in the Overhauser fields the electron subsystem experiences.
