  We present analytical solutions describing a family of both inwardly and
outwardly spiralling orbits in the Kerr spacetime. The solutions are exact, and
remarkable for their simplicity. These orbits all have the angular momentum and
energy of a circular orbit at some radius $r_c$, but are not restricted to
remaining on that circular orbit, a property not possible in Newtonian gravity.
We demonstrate that there are five distinct orbital solutions which terminate
at the black hole singularity, and three solutions which either escape to
infinity or remain bound. The different orbital solutions are characterised
entirely by the black hole spin $a$ and the location of $r_c$. Photon orbits
spiralling into or out of their (unstable) circular orbit radii are also
analysed. These have properties similar to the hyperbolic class of massive
particle orbits discussed herein.
