  We examine the integrability of two models used for the interaction of long
and short waves in dispersive media. One is more classical but arguably cannot
be derived from the underlying water wave equations, while the other one was
recently derived. We use the method of Zakharov and Schulman to attempt to
construct conserved quantities for these systems at different orders in the
magnitude of the solutions. The coupled KdV-NLS model is shown to be
nonintegrable, due to the presence of fourth-order resonances. A coupled real
KdV - complex KdV system is shown to suffer the same fate, except for three
special choices of the coefficients, where higher-order calculations or a
different approach are necessary to conclude integrability or the absence
thereof.
