  We demonstrate existence of positive bound and ground states for a system of
coupled nonlinear Schr\"odinger--Korteweg-de Vries equations. More precisely,
we prove there is a positive radially symmetric ground state if either the
coupling coefficient $\beta>\Lambda$ (for an appropriate constant $\Lambda>0$)
or $\beta>0$ with appropriate conditions on the other parameters of the
problem. Concerning bound states, we prove there exists a positive radially
symmetric bound state if either $0<\beta$ is sufficiently small or
$0<\beta<\Lambda$ with some appropriate conditions on the parameters as for the
ground states. That results give a classification of positive solutions as well
as multiplicity of positive solutions. Furthermore, we consider a system with
more general power nonlinearities, proving the above results, and also we study
natural extended systems with more than two equations. Although the techniques
we employed are variational, we look for critical points of an energy
functional different from the classical one used in this kind of systems. Our
approach improves many of the previous known results, as well as permit us to
show new results not considered or studied before.
