  We continue our study of the well-posedness theory of a one-parameter family
of coupled KdV-type systems in the periodic setting. When the value of a
coupling parameter \alpha \in (0, 4) \setminus 1, we show that the Gibbs
measure is invariant under the flow and the system is globally well-posed
almost surely on the statistical ensemble, provided that certain Diophantine
conditions are satisfied.
