Characterization of Symmetry Properties of First Integrals for Submaximal Linearizable Third-Order ODEs

Table 8

Two symmetry cases and the integrals of (27).

Two symmetries	First integral
𝑋2,𝑋3	𝐹=𝐹(𝐼2)
𝑋3,𝑋4	𝐹=𝐹(𝐼3/𝐼2)
𝑋1,𝑋2+𝑎𝑋3	𝐹=𝐹(𝐼1−(1/2)𝑎𝐼3)
𝑋2+𝑎𝑋3,𝑋4	𝐹=𝐹(𝐼2/(𝐼1−(1/2)𝑎𝐼3))
𝑋1+𝑎𝑋2,𝑋3	𝐹=𝐹(𝐼3−𝑎𝐼2)
𝑋1+𝑎𝑋2,𝑋4	𝐹=𝐹(𝐼3−𝑎𝐼2/𝐼2)
𝑋1+𝑎𝑋3,𝑋2,𝑎≠0	𝐹=𝐹(𝐼1−(1/2)𝑎𝐼2)
𝑋1+𝑎𝑋3,𝑋4,𝑎≠0	𝐹=𝐹(𝐼1−(1/2)𝑎𝐼2/𝐼3)
𝑋2+𝑎𝑋3,𝑋1,𝑎≠0	𝐹=𝐹(𝐼1−(1/2)𝑎𝐼3)
𝑋1+𝑎𝑋2+𝑏𝑋3,𝑋4,𝑎,𝑏≠0	𝐹=𝐹(𝑏𝐼3−2𝑎𝐼1/𝐼2)