  The problem of the anisotropic Inverse Compton scattering between a
monochromatic photon beam and relativistic electrons is revisited and formally
solved without approximations. Solutions are given for the single scattering
with an electron beam and with a population of electrons isotropically
distributed, under the assumption that the energy distribution of the
relativistic particles follows a simple power law as it is the case in many
astrophysical applications. Both the Thomson approximation and the
Klein-Nishina regime are considered for the scattering of an unpolarized photon
beam. The equations are obtained without the ultra-relativistic approximation
and are compared with the ultra-relativistic solutions given in the literature.
The main characteristics of the power distribution and spectra of the scattered
radiation are discussed for relevant examples. In the Thomson case for an
isotropic electron population simple formulae holding down to
mildly-relativistic energies are given. As an application the formulae of the
anisotropic inverse Compton scattering are used to predict the properties of
the X and $\gamma$-ray spectra from the radio lobes of strong FR II radio
galaxies due to the interaction of the relativistic electrons with the incoming
photons from the nucleus. The dependence of the emitted power on the
relativistic electron energy distribution and on its evolution with time is
discussed.
