  We develop a perturbation theory for bright solitons of the F=1 integrable
spinor Bose-Einstein condensate (BEC) model. The formalism is based on using
the Riemann-Hilbert problem and provides the means to analytically calculate
evolution of the soliton parameters. Both rank-one and rank-two soliton
solutions of the model are obtained. We prove equivalence of the rank-one
soliton and the ferromagnetic rank-two soliton. Taking into account a splitting
of a perturbed polar rank-two soliton into two ferromagnetic solitons, it is
sufficient to elaborate a perturbation theory for the rank-one solitons only.
  Treating a small deviation from the integrability condition as a
perturbation, we describe the spinor BEC soliton dynamics in the adiabatic
approximation. It is shown that the soliton is quite robust against such a
perturbation and preserves its velocity, amplitude, and population of different
spin components, only the soliton frequency acquires a small shift. Results of
numerical simulations agree well with the analytical predictions, demonstrating
only slight soliton profile deformation.
