Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability

Table 1

Realizations of three-dimensional Lie algebras in three variables.

Algebra	N	Realization	Rank
𝐴3,1	1	𝜕𝑡,𝜕𝑥,𝜕𝑦	3
	2	𝜕𝑥,𝜕𝑦,𝑡𝜕𝑥+𝑓(𝑡)𝜕𝑦,	2
	3	𝜕𝑥,𝑡𝜕𝑥,𝑦𝜕𝑥	1
	4	𝜕𝑡,𝑥𝜕𝑡,𝜙(𝑥)𝜕𝑡,𝜙^′′(𝑥)≠0	1
𝐴3,2; [𝑋1,𝑋2]=𝑋1	1	𝜕𝑡,𝑡𝜕𝑡+𝜕𝑦,𝜕𝑥	3
	2	𝜕𝑡,𝑡𝜕𝑡+𝑦𝜕𝑥,𝜕𝑥	2
	3	𝜕𝑡,𝑡𝜕𝑡,𝜕𝑥	2
	4	𝜕𝑥,𝑡𝜕𝑡+𝑥𝜕𝑥,𝑡𝜕𝑥	2
𝐴3,3; [𝑋2,𝑋3]=𝑋1	1	𝜕𝑡,𝜕𝑥,𝑥𝜕𝑡	2
	2	𝜕𝑡,𝜕𝑥,𝑥𝜕𝑡+𝑦𝜕𝑥	2
	3	𝜕𝑡,𝜕𝑥,𝑥𝜕𝑡+𝜕𝑦	3
𝐴3,4	1	𝜕𝑡,𝜕𝑥,(𝑡+𝑥)𝜕𝑡+𝑥𝜕𝑥	2
[𝑋1,𝑋3]=𝑋1	2	𝜕𝑡,𝜕𝑥,(𝑡+𝑥)𝜕𝑡+𝑥𝜕𝑥+𝜕𝑦	3
[𝑋2,𝑋3]=𝑋1+𝑋2	3	𝜕𝑥,𝑡𝜕𝑥,−𝜕𝑡+𝑥𝜕𝑥	2
𝐴3,5; [𝑋1,𝑋3]=𝑋1; [𝑋2,𝑋3]=𝑋2	1	𝜕𝑡,𝜕𝑥,𝑡𝜕𝑡+𝑥𝜕𝑥	2
	2	𝜕𝑡,𝜕𝑥,𝑡𝜕𝑡+𝑥𝜕𝑥+𝜕𝑦	3
	3	𝜕𝑥,𝑡𝜕𝑥,𝑥𝜕𝑥	1
	4	𝜕𝑥,𝑡𝜕𝑥,𝑥𝜕𝑥+𝜕𝑦	2
𝐴^𝑎3,6,|𝑎|≤1,𝑎≠0,1	1	𝜕𝑡,𝜕𝑥,𝑡𝜕𝑡+𝑎𝑥𝜕𝑥	2
[𝑋1,𝑋3]=𝑋1	2	𝜕𝑡,𝜕𝑥,𝑡𝜕𝑡+𝑎𝑥𝜕𝑥+𝜕𝑦	3
[𝑋2,𝑋3]=𝑎𝑋2	3	𝜕𝑥,𝑡𝜕𝑥,(1−𝑎)𝑡𝜕𝑡+𝑥𝜕𝑥	2
𝐴^𝑎3,7,𝑎≥0	1	𝜕𝑡,𝜕𝑥,(𝑎𝑡+𝑥)𝜕𝑡+(𝑎𝑥−𝑡)𝜕𝑥	2
[𝑋1,𝑋3]=𝑎𝑋1−𝑋2	2	𝜕𝑡,𝜕𝑥,(𝑎𝑡+𝑥)𝜕𝑡+(𝑎𝑥−𝑡)𝜕𝑥+𝜕𝑦	3
[𝑋2,𝑋3]=𝑋1+𝑎𝑋2	3	𝜕𝑥,𝑡𝜕𝑥,−(1+𝑡^2)𝜕𝑡+(𝑎−𝑡)𝑥𝜕𝑥	2
𝐴3,8,{sl(2,𝑅)}; [𝑋1,𝑋2]=𝑋1; [𝑋2,𝑋3]=𝑋3; [𝑋3,𝑋1]=−2𝑋2	1	𝜕𝑡,𝑡𝜕𝑡,𝑡^2𝜕𝑡	1
	2	𝜕𝑡,𝑡𝜕𝑡+𝑥𝜕𝑥,𝑡^2𝜕𝑡+2𝑡𝑥𝜕𝑥+𝑥𝜕𝑦	3
	3	𝜕𝑡+𝜕𝑥,𝑡𝜕𝑡+𝑥𝜕𝑥,𝑡^2𝜕𝑡+𝑥^2𝜕𝑥	2
	4	−𝑡𝜕𝑥,12(−𝑡𝜕𝑡+𝑥𝜕𝑥),𝑥𝜕𝑡	2
	5	𝜕𝑡,𝑡𝜕𝑡+𝑥𝜕𝑥,(𝑡^2−𝑥^2)𝜕𝑡+2𝑡𝑥𝜕𝑥	2
𝐴3,9,{so(3)}; [𝑋1,𝑋2]=𝑋3	1	(1+𝑡^2)𝜕𝑡+𝑥𝑡𝜕𝑥,𝑥𝜕𝑡−𝑡𝜕𝑥,−𝑥𝑡𝜕𝑡−(1+𝑥^2)𝜕𝑥	2
[𝑋3,𝑋1]=𝑋2; [𝑋2,𝑋3]=𝑋1	2	−sin𝑡tan𝑥𝜕𝑡−cos𝑡𝜕𝑥+sin𝑡sec𝑥𝜕𝑦,−cos𝑡tan𝑥𝜕𝑡+sin𝑡𝜕𝑥+cos𝑡sec𝑥𝜕𝑦,𝜕𝑡	3