The Cauchy problem and integrability of a modified Euler-Poisson equation

We prove that the periodic initial value problem for a modified Euler-Poisson equation is well-posed for initial data in when . We also study the analytic regularity of this problem and prove a Cauchy-Kowalevski type theorem. After presenting a formal derivation of the equation on the semidirect product space as a Hamiltonian equation, we concentrate on one space dimension () and show that the equation is bihamiltonian.