A Note on Four-Dimensional Symmetry Algebras and Fourth-Order Ordinary Differential Equations

Table 1

Lie algebra	𝑁	Realizations (generators)
4𝐴1	11	𝜕𝑡,𝑥𝜕𝑡,𝑘(𝑥)𝜕𝑡,ℎ(𝑥)𝜕𝑡
2𝐴2.1	5	𝜕𝑡,𝑡𝜕𝑡,𝜕𝑥,𝑥𝜕𝑥
	*7	𝜕𝑡,𝑡𝜕𝑡+𝑥𝜕𝑥,𝑥𝜕𝑡,−𝑥𝜕𝑥
𝐴3.2⊕𝐴1	9	𝜕𝑡,𝑥𝜕𝑡,𝑡𝜕𝑡−𝜕𝑥,𝑒^−𝑥𝜕𝑡
𝐴^𝑎3.4⊕𝐴1,|𝑎|≤1,𝑎≠0,1	9	𝜕𝑡,𝑥𝜕𝑡,𝑡𝜕𝑡+(1−𝑎)𝑥𝜕𝑥,|𝑥|^1/(1−𝑎)𝜕𝑡
𝐴^𝑏3.5⊕𝐴1,𝑏≥0	8	𝜕𝑡,𝑥𝜕𝑡,(𝑏−𝑥)𝑡𝜕𝑡−(1+𝑥^2)𝜕𝑥,√1+𝑥^2𝑒^−𝑏arctan𝑥𝜕𝑡
Sl(2,ℝ)⊕𝐴1	*8	𝜕𝑡,𝑡𝜕𝑡+𝑥𝜕𝑥,𝑡^2𝜕𝑡+2𝑡𝑥𝜕𝑥,𝑥𝜕𝑥
	9	𝜕𝑡,𝑡𝜕𝑡,𝑡^2𝜕𝑡,𝜕𝑥
𝐴4.1	8	𝜕𝑡,𝑥𝜕𝑡,(1/2)𝑥^2𝜕𝑡,−𝜕𝑥
𝐴^𝑏4.2,𝑏≠0	8	𝜕𝑡,𝑥𝜕𝑡,(1/(1−𝑏))𝑥ln|𝑥|𝜕𝑡,𝑏𝑡𝜕𝑡+(𝑏−1)𝑥𝜕𝑥, 𝑏≠1
𝐴4.3	8	𝜕𝑡,𝑥𝜕𝑡,−𝑥ln|𝑥|𝜕𝑡,𝑡𝜕𝑡+𝑥𝜕𝑥
𝐴4.4	7	𝜕𝑡,𝑥𝜕𝑡,(1/2)𝑥^2𝜕𝑡,𝑡𝜕𝑡−𝜕𝑥
𝐴^𝑎,𝑏,𝑐4.5,−1≤𝑎<𝑏<𝑐=1,𝑎𝑏𝑐≠0	7	𝜕𝑡,𝑒^(𝑎−𝑏)𝑥𝜕𝑡,𝑒^(𝑎−1)𝑥𝜕𝑡,𝑎𝑡𝜕𝑡+𝜕𝑥, 𝑏>0 if 𝑎=−1
	10	𝜕𝑡,𝑥𝜕𝑡,𝜙(𝑥)𝜕𝑡,𝑡𝜕𝑡, 𝜙^′′(𝑥)≠0,𝑎=𝑏=𝑐=1
𝐴^𝑎,𝑏4.6,𝑎>0	6	𝜕𝑡,𝑒^(𝑎−𝑏)𝑥cos𝑥𝜕𝑡,−𝑒^(𝑎−𝑏)𝑥sin𝑥𝜕𝑡,𝑎𝑡𝜕𝑡+𝜕𝑥
𝐴4.7	5	𝜕𝑡,𝑥𝜕𝑡,−𝜕𝑥,(2𝑡−(1/2)𝑥^2)𝜕𝑡+𝑥𝜕𝑥
𝐴^𝑏4.8,|𝑏|≤1	5	𝜕𝑡,𝜕𝑥,𝑥𝜕𝑡,(1+𝑏)𝑡𝜕𝑡+𝑥𝜕𝑥
	*7	𝜕𝑡,𝑥𝜕𝑡,−𝜕𝑥,(1+𝑏)𝑡𝜕𝑡+𝑏𝑥𝜕𝑥, 𝑏≠±1
𝐴4.10	*6	𝜕𝑡,𝜕𝑥,𝑡𝜕𝑡+𝑥𝜕𝑥,𝑥𝜕𝑡−𝑡𝜕𝑥
	7	𝜕𝑡,𝑥𝜕𝑡,𝑡𝜕𝑡,−𝑡𝑥𝜕𝑡−(1+𝑥^2)𝜕𝑥