In this work, our goal is to study the Cauchy problem for some generalization of the following system. iφt + ∆φ = −div(|∇ψ|2∇ψ), ∆ψ = φ. In particular we shall construct solutions in the energy space associ- ated to this system. We give some sufficient conditions on the initial data which ensure that the solutions are global, but we show that in some cases, finite time blow up occures.