New exact solutions of the Navier-Stokes equations are obtained for steady-state three-dimensional conical flows. In this class of flows the velocity decreases in inverse proportion to the distance from the source and the input equations reduce to two-dimensional ones. It is shown that in the spherical coordinate system the equations of motion reduce to a single nonlinear equation with respect to a scalar function which depends on the polar angles. The case in which this equation reduces to the integrable Liouville equation is discussed. This makes it possible to obtain a wide class of three-dimensional solutions in analytic form.