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

Table 4

Two symmetry cases and the integrals of (1).

Two symmetries	First integral
𝑋1, 𝑋2	𝐹=𝐹(𝐼2𝐼3)
𝑋1, 𝑋5	𝐹=𝐹(𝐼1(𝐼2𝐼3)^−1/2)
𝑋2, 𝑋5	𝐹=𝐹(𝐼3/𝐼2)
𝑋2±𝑋4,𝑋5	𝐹=𝐹(𝐼3/(𝐼2±𝐼1))
𝑋3,𝑋5	𝐹=𝐹(𝐼2/𝐼1)
𝑋5+𝑎𝑋1,𝑋2,𝑎≠0	𝐹=𝐹(𝐼3𝐼2^𝑎−1/𝑎+1)
𝑋5+𝑎𝑋1,𝑋3	𝐹=𝐹(𝐼2𝐼1^−1−𝑎)
𝑋2,𝑋1+𝑋5+𝑋3	𝐹=𝐹(𝐼3−ln𝐼2)
𝑋4,𝑋1+𝑋5+𝑋3	𝐹=𝐹(𝐼3−ln𝐼^21)