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

Table 2

Lie algebra	𝑁	Realizations and equations
2𝐴2.1	*7	𝜕𝑡,𝑡𝜕𝑡+𝑥𝜕𝑥,𝑥𝜕𝑡,−𝑥𝜕𝑥
		^⋅𝑥^⋅⋅⋅=10𝑥̈𝑥⃛𝑥̇−15𝑥̈^3𝑥̇^2+𝑥̇^2𝑥̈𝑥^2𝑓𝑥𝑥⃛𝑥̇𝑥̈−3𝑥𝑥̈𝑥̇^2
Sl(2,ℝ)⊕𝐴1	*8	𝜕𝑡,𝑡𝜕𝑡+𝑥𝜕𝑥,𝑡^2𝜕𝑡+2𝑡𝑥𝜕𝑥,𝑥𝜕𝑥
		^⋅𝑥^⋅⋅⋅2𝑥̇𝑥⃛=−𝑥+1𝑥^3𝑥̇^22−𝑥𝑥̈^2𝑓𝑥^2𝑥⃛(𝑥̇^2/2−𝑥𝑥̈)^3/2
𝐴^𝑏4.8,|𝑏|≤1	*7	𝜕𝑡,𝑥𝜕𝑡,−𝜕𝑥,(1+𝑏)𝑡𝜕𝑡+𝑏𝑥𝜕𝑥, 𝑏≠±1
		^⋅𝑥^⋅⋅⋅=10𝑥̈𝑥⃛𝑥̇−15𝑥̈^3𝑥̇^2+𝑥̈^(1−3𝑏)/(1−𝑏)𝑥̇^(4−2𝑏)/(1−𝑏)𝑓𝑥̈𝑥̇^4−3𝑥̈^2𝑥̇^5𝑥̈𝑥̇^3(1−2𝑏)/(𝑏−1)
𝐴4.10	*6	𝜕𝑡,𝜕𝑥,𝑡𝜕𝑡+𝑥𝜕𝑥,𝑥𝜕𝑡−𝑡𝜕𝑥
		^⋅𝑥^⋅⋅⋅=10𝑥̇𝑥̈𝑥⃛(1+𝑥̇^2)−15𝑥̇^2𝑥̈^3(1+𝑥̇^2)^2+𝑥̈^3(1+𝑥̇^2)^2𝑓1+𝑥̇^2𝑥⃛𝑥̈^2−3𝑥̇