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

Table 7

One symmetry cases and the integrals of (27).

One symmetry	First integral
𝑋3	𝐹=𝐹(𝐼2,𝐼3)
𝑋4	𝐹=𝐹(𝐼2/𝐼1,𝐼3/𝐼2)
𝑋2+𝑎𝑋3	𝐹=𝐹(𝐼2𝐼3,𝐼2exp(𝐼1))
𝑋2±𝑋4	𝐹=𝐹(𝐼1−(1/2)𝑎𝐼3,𝐼2)
𝑋1+𝑎𝑋2	𝐹=𝐹(𝐼3−𝑎𝐼2,𝐼1)
𝑋1+𝑎𝑋2	𝐹=𝐹(𝐼1−(1/2)𝑎𝐼2,𝐼3)
𝑋1+𝑎𝑋2+𝑏𝑋3,𝑎,𝑏≠0	𝐹=𝐹(𝑏𝐼3−2𝑎𝐼1,𝐼2)