  The paper aims at developing the Riemann-Hilbert (RH) approach for the
modified Camassa-Holm (mCH) equation on the line with non-zero boundary
conditions, in the case when the solution is assumed to approach two different
constants at different sides of the line. We present detailed properties of
spectral functions associated with the initial data for the Cauchy problem for
the mCH equation and obtain a representation for the solution of this problem
in terms of the solution of an associated RH problem.
