  We introduce a parametric coupled KdV system which contains, for particular
values of the parameter, the complex extension of the KdV equation and one of
the Hirota-Satsuma integrable systems. We obtain a generalized Gardner
transformation and from the associated $\varepsilon$- deformed system we get
the infinite sequence of conserved quantities for the parametric coupled
system. We also obtain a B\"{a}cklund transformation for the system. We prove
the associated permutability theorem corresponding to such transformation and
we generate new multi-solitonic and periodic solutions for the system depending
on several parameters. We show that for a wide range of the parameters the
solutions obtained from the permutability theorem are regular solutions.
Finally we found new multisolitonic solutions propagating on a non-trivial
regular static background.
