  The paper is devoted to the study of the short rate equation of the form $$
dR(t)=F(R(t))dt+\sum_{i=1}^{d}G_i(R(t-))dZ_i(t), \quad R(0)=x\geq 0, \quad t>0,
$$ with deterministic functions $F,G_1,...,G_d$ and independent L\'evy
processes of infinite variation $Z_1,...,Z_d$ with regularly varying Laplace
exponents. The equation is supposed to have a nonnegative solution which
generates an affine term structure model. A precise form of the generator of
$R$ is characterized and a related classification of equations which generate
affine models introduced in the spirit of Dai and Singleton
\cite{DaiSingleton}. Each class is shown to have its own canonical
representation which is an equation with the same drift and the jump diffusion
part based on a L\'evy process taking values in $\mathbb{R}^{g}, 1\leq g\leq
d$, with independent coordinates being stable processes with stability indices
in the range $(1,2]$. Numerical calibration results of canonical
representations to the market term structure of interest rates are presented
and compared with the classical CIR model. The paper generalizes the classical
results on the CIR model from \cite{CIR}, as well as on its extended version
from \cite{BarskiZabczykCIR} and \cite{BarskiZabczyk} where $Z$ was a
one-dimensional L\'evy process.
