  A new technique is proposed for the solution of the Riemann-Hilbert problem
with the Chebotarev-Khrapkov matrix coefficient
$G(t)=\alpha_1(t)I+\alpha_2(t)Q(t)$, $\alpha_1(t), \alpha_2(t)\in H(L)$, $Q(t)$
is a $2\times 2$ zero-trace polynomial matrix, and $I$ is the unit matrix. This
problem has numerous applications in elasticity and diffraction theory. The
main feature of the method is the removal of the essential singularities of the
solution to the associated homogeneous scalar Riemann-Hilbert problem on the
hyperelliptic surface of an algebraic function by means of the Baker-Akhiezer
function. The consequent application of this function for the derivation of the
general solution to the vector Riemann-Hilbert problem requires finding of the
$\rho$ zeros of the Baker-Akhiezer function ($\rho$ is the genus of the
surface). These zeros are recovered through the solution to the associated
Jacobi problem of inversion of abelian integrals or, equivalently, the
determination of the zeros of the associated degree-$\rho$ polynomial and
solution of a certain linear algebraic system of $\rho$ equations.
