  Future observations of the Sunyaev-Zeldovich (SZ) effect promise ever
improving measurements in terms of both sensitivity and angular resolution. As
such, it is increasingly relevant to model `higher-order' contributions to the
SZ effect. This work examines the effects of high-energy non-thermal electron
distributions and those of anisotropic electron and photon distributions on the
SZ signals. Analytic forms of the anisotropic scattering kernels for photons
and electrons have been derived and investigated. We present a method for
determining the anisotropic contributions through a spherical harmonic
decomposition to arbitrary angular multipoles, and discuss the behaviour of
these scattering kernels. We then carry out an exploration of various
simplistic models of high energy non-thermal electron distributions, and
examine their anisotropic behaviour. The kinematic SZ in the relativistic
regime is studied using the kernel formulation allowing us to clarifying the
role of kinematic corrections to the scattering optical depth. We finally
present a release of an updated and refined version of SZpack including a new
integrated Python interface and new modules for the calculation of various SZ
signals, including those described in this paper.
