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

Table 2

The adjoint table for the symmetries (2).

Ad	𝑋1	𝑋2	𝑋3	𝑋4	𝑋5
𝑋1	𝑋1	𝑋2	𝑒^−𝜖𝑋3	𝑒^𝜖𝑋4	𝑋5
𝑋2	𝑋1	𝑋2	𝑋3	𝑋4	𝑋5−𝜖𝑋2
𝑋3	𝑋1+𝜖𝑋3	𝑋2	𝑋3	𝑋4	𝑋5−𝜖𝑋3
𝑋4	𝑋1−𝜖𝑋4	𝑋2	𝑋3	𝑋4	𝑋5−𝜖𝑋4
𝑋5	𝑋1	𝑒^𝜖𝑋2	𝑒^𝜖𝑋3	𝑒^𝜖𝑋4	𝑋5