Generalized Emden–Fowler equations in noncentral curl forces and first integrals

Noncentral force is an exemplary example of curl force and in general this is not an integrable system. The purpose of this note is twofold. First, we study a different reduction in the noncentral force compared to Berry and Shukla, and this leads to the generalized Emden–Fowler (GEF) equation, which in turn can be mapped to the Thomas–Fermi equation. Second, we compute the first integrals of the integrable standard Emden–Fowler (EF) and the generalized EF equations associated with the reduced noncentral dynamics using old results and new techniques. Finally, we compute the reduction in the nonpolynomial noncentral forces, which also leads to generalized EF equations.