  We consider the local well-posedness problem of a one-parameter family of
coupled KdV-type systems both in the periodic and non-periodic setting. In
particular, we show that certain resonances occur, closely depending on the
value of a coupling parameter \alpha when \alpha \ne 1. In the periodic
setting, we use the Diophantine conditions to characterize the resonances, and
establish sharp local well-posedness of the system in H^s(\mathbb{T}_\lambda),
s \geq s^\ast, where s^\ast = s^\ast(\alpha) \in ({1/2}, 1] is determined by
the Diophantine characterization of certain constants derived from the coupling
parameter \alpha. We also present a sharp local (and global) result in
L^2(\mathbb{R}). In the appendix, we briefly discuss the local well-posedness
result in H^{-{1/2}}(\mathbb{T}_\lambda) for \alpha= 1 without the mean 0
assumption, by introducing the vector-valued X^{s, b} spaces.
