It is shown by solving an equation for the vector potential A(r, z) of the magnetic field that under appropriate conditions the focusing properties of a betatron magnetic field are periodic with respect to the z coordinate. Under these conditions there may be several equilibrium orbits lying in parallel planes z equals O, z equals z//0//1 . . . , z equals mz//0//1 in the accelerator. An equation is derived for the distance z//0//1 between the equilbrium orbit planes for a given orbit radius r//0 and field decay exponent n//0. The operation of such accelerators is described.