  For the case of nonlinear second-order differential equations with a constant
coefficient of the first derivative term and polynomial nonlinearities, the
factorization conditions of Rosu and Cornejo-Perez are approached in two ways:
(i) by commuting the subindices of the factorization functions in the two
factorization conditions and (ii) by leaving invariant only the first
factorization condition achieved by using monomials or polynomial sequences.
For the first case, the factorization brackets commute and the generated
equations are only equations of Ermakov-Pinney type. The second modification is
non commuting, leading to nonlinear equations with different nonlinear force
terms, but the same first-order part as the initially factored equation. It is
illustrated for monomials with the examples of the generalized Fisher and
FitzHugh-Nagumo initial equations. A polynomial sequence example is also
included.
