Based on \cite{geier2015cumulant}, collision operator  $\vec{C}$:
\begin{equation*}
\vec{C}\left( \vec{f} \right) = \matr{M}^{-1} \vec{G}^{-1}\left(\matr{N}^{-1} \matr{S}\matr{N}\left(\vec{\kum}^{(eq)} - \vec{G}\left(\matr{M}\vec{f}\right) \right)\right),
\end{equation*}
where
\input{\path/d3q27_cum_matS2}
$\oA,\oB,\cdots, \oJ \in \left(0,2\right)$ 
and matrix $\matr{N}$ defines combination of cumulants for the collision by
\input{\path/d3q27_cum_matN2}

The nonlinear operator $\vec{G}$ is the same as in CuLBM1 in Section~\ref{section:CuLBM1} and, again,
%transforms the raw moment vector $\vec{\mu}$ defined by the matrix $\matr{M}$ to the cumulant vector 
%$$
%	\vec{\kum} = \vec{G}(\vec{\mu}) = \left( \kum_{(0,0,0)}, \kum_{(1,0,0)}, \kum_{(0,1,0)}, \cdots, \kum_{(2,2,2)}\right)^T
%$$ 
%as in CuLBM1 (Section~\ref{section:CuLBM1}).
%\input{\path/d3q27_cum_kappa}
%and $\vec{G}^{-1}$ is its inverse.
the equilibrium cumulant vector $\vec{\kum}^{(eq)}$ is defined by
\input{\path/d3q27_cum2_eq}

