We return to the representation of solutions (u's) to Eq(3) in the previous section (2.2).
We first show an isomorphism between finding polynomial first integrals u's, and determining the simultaneous invariants and covariants of a collection of binary forms [Hil93]. This relation allows us to use the results from invariant theory and methods from computational commutative algebra, namely Gröbner bases [CLO97], to show that there is a finite basis {uB} for the ring {u}, and every u can be written as a polynomial in this finite basis. Also, we can compute such a basis for specific cases. Moreover, we describe the Hironaka decomposition [Stu93] of the ring {u} that yields unique representations for u.
This method gives an alternate derivation of the normal forms and, in particular, a representation of all the polynomial solutions to Eq(2).