The dynamic interaction of N symmetric pairs of point vortices with a neutrally buoyant two-dimensional rigid circular cylinder in the inviscid Hamiltonian model of Shashikanth et al. ͓Phys. Fluids 14, 1214 ͑2002͔͒ and Shashikanth ͓Reg. Chaotic Dyn. 10, 1 ͑2005͔͒ is examined. The model may be thought of as a section of an inviscid axisymmetric model of a neutrally buoyant sphere interacting with N coaxial circular vortex rings and has possible applications to problems such as fish swimming. The Hamiltonian structure of this half-space model is first presented. The cases N = 1 and N = 2 are then examined in detail. Equilibria and bifurcations are studied, and for both these cases an important bifurcation parameter involving the total linear "momentum" of the system, the strength of the vortex pairs, and the radius of the cylinder emerges. For N = 1, it is shown that there exist the moving Föppl equilibrium and the moving normal line equilibrium ͑in which the vortices in the pair are located on the top and bottom of the moving cylinder͒. For N = 2, when ⌫ 1 = ⌫ 2 and when ⌫ 1 =−⌫ 2 , there exists another set of equilibrium configurations. Linear stability analysis of all these equilibria, within the symmetric class of solutions, is carried out and phase portraits presented. In addition, for N = 1, the velocity and acceleration surfaces for the cylinder are presented.