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

Table 3

One symmetry cases and the integrals of (1).

One symmetry	First integral
𝑋1	𝐹=𝐹(𝐼1,𝐼2𝐼3)
𝑋2	𝐹=𝐹(𝐼2,𝐼3)
𝑋1+𝑋2	𝐹=𝐹(𝐼2𝐼3,𝐼2exp(𝐼1))
𝑋2±𝑋4	𝐹=𝐹(𝐼2±𝐼1,𝐼3)
𝑋5	𝐹=𝐹(𝐼3/(𝐼2±𝐼1))
𝑋5+𝑎𝑋1,𝑎≠0	𝐹=𝐹(𝐼2𝐼1^−1−𝑎,𝐼3𝐼1^𝑎−1)
𝑋1+𝑋5+𝑋3	𝐹=𝐹(𝐼2𝐼1^−2,𝐼3−ln𝐼2)