Symmetry reductions of the $(3+1)$ -dimensional modified Zakharov–Kuznetsov equation

Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 1 Commutation relation of the Lie algebra of Eq. (2)

	$\mathcal{V}_{1}$	$\mathcal{V}_{2}$	$\mathcal{V}_{3}$	$\mathcal{V}_{4}$	$\mathcal{V}_{5}$	$\mathcal{V}_{6}$
$\mathcal{V}_{1}$	0	$\frac{2}{3}k\mathcal{V}_{3}-\mathcal{V}_{2}$	$-\frac{1}{3}\mathcal{V}_{3}$	0	$-\frac{1}{3}\mathcal{V}_{5}$	$-\frac{1}{3}\mathcal{V}_{6}$
$\mathcal{V}_{2}$	$\mathcal{V}_{2}-\frac{2}{3}k\mathcal{V}_{3}$	0	0	0	0	0
$\mathcal{V}_{3}$	$\frac{1}{3}\mathcal{V}_{3}$	0	0	0	0	0
$\mathcal{V}_{4}$	0	0	0	0	$\mathcal{V}_{6}$	$-\mathcal{V}_{5}$
$\mathcal{V}_{5}$	$\frac{1}{3}\mathcal{V}_{5}$	0	0	$-\mathcal{V}_{6}$	0	0
$\mathcal{V}_{6}$	$\frac{1}{3}\mathcal{V}_{6}$	0	0	$\mathcal{V}_{5}$	0	0

	\(\mathcal{V}_{1}\)	\(\mathcal{V}_{2}\)	\(\mathcal{V}_{3}\)	\(\mathcal{V}_{4}\)	\(\mathcal{V}_{5}\)	\(\mathcal{V}_{6}\)
\(\mathcal{V}_{1}\)	0	\(\frac{2}{3}k\mathcal{V}_{3}-\mathcal{V}_{2}\)	\(-\frac{1}{3}\mathcal{V}_{3}\)	0	\(-\frac{1}{3}\mathcal{V}_{5}\)	\(-\frac{1}{3}\mathcal{V}_{6}\)
\(\mathcal{V}_{2}\)	\(\mathcal{V}_{2}-\frac{2}{3}k\mathcal{V}_{3}\)	0	0	0	0	0
\(\mathcal{V}_{3}\)	\(\frac{1}{3}\mathcal{V}_{3}\)	0	0	0	0	0
\(\mathcal{V}_{4}\)	0	0	0	0	\(\mathcal{V}_{6}\)	\(-\mathcal{V}_{5}\)
\(\mathcal{V}_{5}\)	\(\frac{1}{3}\mathcal{V}_{5}\)	0	0	\(-\mathcal{V}_{6}\)	0	0
\(\mathcal{V}_{6}\)	\(\frac{1}{3}\mathcal{V}_{6}\)	0	0	\(\mathcal{V}_{5}\)	0	0