  We analyze the properties of a polytropic fluid which is radially accreted
into a Schwarzschild black hole. The case where the adiabatic index gamma lies
in the range 1 < gamma <= 5/3 has been treated in previous work. In this
article we analyze the complementary range 5/3 < gamma <= 2. To this purpose,
the problem is cast into an appropriate Hamiltonian dynamical system whose
phase flow is analyzed. While for 1 < gamma <= 5/3 the solutions are always
characterized by the presence of a unique critical saddle point, we show that
when 5/3 < gamma <= 2, an additional critical point might appear which is a
center point. For the parametrization used in this paper we prove that whenever
this additional critical point appears, there is a homoclinic orbit.
