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We present an exact $γ=5/3$ spherical accretion solution which modifies the Bondi boundary condition of $ρ\to const.$ as $r\to \infty$ to $ρ\to 0$ as $r \to \infty$. This change allows for simple power law solutions on the density and infall velocity fields, ranging from a cold empty free-fall condition where pressure tends to zero, to a hot hydrostatic equilibrium limit with no infall velocity. As in the case of the Bondi solution, a maximum accretion rate appears. As in the $γ=5/3$ case of the Bondi solution, no sonic radius appears, this time however, because the flow is always characterised by a constant Mach number. This number equals 1 for the case of the maximum accretion rate, diverges towards the cold empty state, and becomes subsonic towards the hydrostatic equilibrium limit. It can be shown that in the limit as { $r \to 0$}, the Bondi solution tends to the new solution presented, { extending the validity of the Bondi accretion value to} cases where the accretion density profile does not remain at a fixed constant value out to infinity. We then explore small deviations from sphericity and the presence of angular momentum through an analytic perturbative analysis. Such perturbed solutions yield a rich phenomenology through density and velocity fields in terms of Legendre polynomials, which we begin to explore for simple angular velocity boundary conditions having zeros on the plane and pole. The new solution presented provides complementary physical insight into accretion problems in general.