  We present the hierarchy and soliton solutions associated to a
multi-component generalisation of the modified Korteweg-de Vries equation. A
recursive formula for obtaining the Lax operators associated to the higher
flows of the hierarchy is provided. Using the method of rational dressing and
the symmetries of the Lax operators we obtain the one-soliton solution. We also
derive the general rank one-breather solution and express it in terms of
determinants. We present the simplest case of the one-breather solution, which
is parametrised by two constant unit vectors normal to each other. Finally, we
obtain the associated B\"acklund transformation for the hierarchy.
