  We study a one-dimensional Bose gas with two internal states described by the
Yang-Gaudin model and calculate analytically the dispersion relation of a
polaron quasiparticle, which is the lowest excitation branch. We study the
dispersion in the thermodynamic limit in the regimes of weak and strong
interaction without limitations on the momentum. At weak interaction the
polaron dispersion is in the vicinity of the dark soliton one; we calculate the
leading deviation in the parametric form. At strong interaction we find an
ansatz for the explicit form of the polaron dispersion. It has the form of a
power series of the sine function of the momentum with interaction-dependent
coefficients. By increasing the power of the series, the corresponding
coefficients show faster decay and thus one practically needs only a few of
them; we give the results for the first three. The coefficients of the series
are connected to the Maclaurin series of the polaron dispersion and thus it is
sufficient to calculate latter quantity to automatically find the power series
result for the dispersion at all momenta. The derived results for the
dispersion can be used to obtain explicit expressions for the exponents of the
power-law singularities in the response functions at the spectral edge.
