  The matrix 2x2 spectral differential equation of the second order is
considered on x in ($-\infty,+\infty$). We establish elementary Darboux
transformations covariance of the problem and analyze its combinations. We
select a second covariant equation to form Lax pair of a coupled KdV-MKdV
system. The sequence of the elementary Darboux transformations of the
zero-potential seed produce two-parameter solution for the coupled KdV-MKdV
system with reductions. We show effects of parameters on the resulting
solutions (reality, singularity). A numerical method for general coupled
KdV-MKdV system is introduced. The method is based on a difference scheme for
Cauchy problems for arbitrary number of equations with constants coefficients.
We analyze stability and prove the convergence of the scheme which is also
tested by numerical simulation of the explicit solutions.
