  We introduce an exactly integrable nonlinear model describing the dynamics of
spinor solitons in space-dependent matrix gauge potentials of rather general
types. The model is shown to be gauge equivalent to the integrable system of
vector nonlinear Schr\"odinger equations known as the Manakov model. As an
example we consider a self-attractive Bose-Einstein condensate with random
spin-orbit coupling (SOC). If Zeeman splitting is also included, the system
becomes nonintegrable. We illustrate this by considering the random walk of a
soliton in a disordered SOC landscape. While at zero Zeeman splitting the
soliton moves without scattering along linear trajectories in the random SOC
landscape, at nonzero splitting it exhibits strong scattering by the SOC
inhomogeneities. For a large Zeeman splitting the integrability is recovered.
In this sense the Zeeman splitting serves as a parameter controlling the
crossover between two different integrable limits.
