  The study of generic, non-linear, deformations of Special Relativity
parametrized by a high-energy scale $M$, which was carried out at first order
in $M$ in Phys.Rev. D86, 084032 (2012), is extended to second order. This can
be done systematically through a ('generalized') change of variables from
momentum variables that transform linearly. We discuss the different
perspectives on the meaning of the change of variables, obtain the coefficients
of modified composition laws and Lorentz transformations at second order, and
work out how $\kappa$-Poincar\'e, the most commonly used example in the
literature, is reproduced as a particular case of the generic framework exposed
here.
