Algebraic Properties of First Integrals for Scalar Linear Third-Order ODEs of Maximal Symmetry

Table 1

Realizations of three-dimensional algebras in the real plane.

𝑝=𝜕/𝜕𝑥 and 𝑞=𝜕/𝜕𝑦
Algebra	Realizations in (𝑥, 𝑦) plane
𝐿3;1	𝑋1=𝑞, 𝑋2=𝑥𝑞, 𝑋3=𝑥^2𝑞
𝐿3;2	𝑋1=𝑞, 𝑋2=𝑝, 𝑋3=𝑥𝑞
𝐿^I3;3	𝑋1=𝑞, 𝑋2=𝑝, 𝑋3=𝑥𝑝+(𝑥+𝑦)𝑞
𝐿^II3;3	𝑋1=𝑞, 𝑋2=𝑥𝑞, 𝑋3=𝑝+𝑦𝑞
𝐿^I3;4	𝑋1=𝑝, 𝑋2=𝑞, 𝑋3=𝑥𝑝
𝐿^II3;4	𝑋1=𝑞, 𝑋2=𝑥𝑞, 𝑋3=𝑥𝑝+𝑦𝑞
𝐿^I3;5	𝑋1=𝑝, 𝑋2=𝑞, 𝑋3=𝑥𝑝+𝑦𝑞
𝐿^II3;5	𝑋1=𝑞, 𝑋2=𝑥𝑞, 𝑋3=𝑦𝑞
𝐿^I3;6	𝑋1=𝑝, 𝑋2=𝑞, 𝑋3=𝑥𝑝+𝑎𝑦𝑞, 𝑎≠0,1
𝐿^II3;6	𝑋1=𝑞, 𝑋2=𝑥𝑞, 𝑋3=(1−𝑎)𝑥𝑝+𝑦𝑞, 𝑎≠0,1
𝐿^I3;7	𝑋1=𝑝, 𝑋2=𝑞, 𝑋3=(𝑏𝑥+𝑦)𝑝+(𝑏𝑦−𝑥)𝑞
𝐿^II3;7	𝑋1=𝑥𝑞, 𝑋2=𝑞, 𝑋3=(1+𝑥^2)𝑝+(𝑥+𝑏)𝑦𝑞
𝐿^I3;8	𝑋1=𝑝, 𝑋2=𝑥𝑝+𝑦𝑞, 𝑋3=𝑥^2𝑝+2𝑥𝑦𝑞
𝐿^II3;8	𝑋1=𝑞, 𝑋2=𝑥𝑝+𝑦𝑞, 𝑋3=2𝑥𝑦𝑝+(𝑦^2−𝑥^2)𝑞
𝐿^III3;8	𝑋1=𝑞, 𝑋2=𝑥𝑝+𝑦𝑞, 𝑋3=2𝑥𝑦𝑝+(𝑦^2+𝑥^2)𝑞
𝐿^IV3;8	𝑋1=𝑞, 𝑋2=𝑦𝑞, 𝑋3=𝑦^2𝑞
𝐿3;9	𝑋1=(1+𝑥^2)𝑝+𝑥𝑦𝑞, 𝑋2=𝑥𝑦𝑝+(1+𝑦^2)𝑞,
	𝑋3=𝑦𝑝−𝑥𝑞