  We prove that the periodic initial value problem for a modified Euler-Poisson
equation is well-posed for initial data in $H^{s} (T^{m})$ when $s>m/2+2$ and
we improve the Sobolev index to $s>3/2$ for $m=1$. 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
$ Diff \ltimes C^{\infty}(\tor)$ as a Hamiltonian equation, we concentrate to
one space dimension ($m=1$) and show that the equation is bihamiltonian.
