The problem of the motion of a solid in a flow of particles around a f
ixed point is considered. This problem is well known to have an extrem
ely non-conservative form. Nevertheless, it turns out that the dynamic
s of the solid in this problem can be described, with certain assumpti
ons, by a system of Hamiltonian equations. The conditions under which
this quite unexpected fact can occur, are investigated. The presence o
f a Hamiltonian structure considerably increases the interest in the p
roblem of the existence of additional first integrals of the equations
of motion. It turns out that there are certain cases when these integ
rals exist. These include the case when the equations of motion allow
the possibility of an integral similar to a Hess integral in the probl
em of the motion of a heavy solid around a fixed point. The steady-sta
te motions of the system considered are also determined, and their sta
bility is investigated.