  We analyse a model of a one-dimensional electron gas interacting with an
antiferromagnetic Heisenberg spin-1/2 chain via the spin exchange interactions.
Using a solution at a special limit, we characterize the gapless modes of the
spin gap fixed point at weak coupling $ J_{K}\ll J_{H},E_{F} $. we show that
the only gapless pairing mode with divergent susceptibility is a composite
odd-parity odd-frequency singlet pairing order parameter, while the ordinary
BCS even-parity singlet pairing mode is incoherent. For 2-leg ladder systems,
we note that it is possible to have a range of doping where the chemical
potential cuts only the anti-bonding band while the bonding band remains
half-filled. We propose that, in such a state, the 2-leg ladder is effectively
realizing the one-dimensional Kondo-Heisenberg model.
