In the search for a more general recipe, however, we choose to eliminate the first-order derivatives Z $^{′}$( z ), Φ $^{′}$( ϕ ), and R $^{′}$( ρ ). At this point, the elimi- nation of the first-order derivatives of Z ( z ) and Φ ( ϕ ) is achieved through the substitutions to imply that and Whereas, the elimination of the first-order derivative of R ( ρ ) may be sought through the substitutions (with the restriction that b is a non-zero constant to avoid triviality) to imply that Z ( z ) = √ k ( z ) ˜ Z ( z ) and Φ ( ϕ ) = √ f ( ϕ ) ˜ Φ ( ϕ ) , (6) Z $^{′′}$( z ) Z ( z ) − k $^{′}$( z ) k ( z ) Z $^{′}$( z ) Z ( z ) = − 3 4 ( k $^{′}$( z ) k ( z ) ) 2 + 1 2 k ′′ ( z ) k ( z ) + ˜ Z $^{′′}$( z ) ˜ Z ( z ) (7) Φ $^{′′}$( ϕ ) Φ ( ϕ ) − f $^{′}$( ϕ ) f ( ϕ ) Φ $^{′}$( ϕ ) Φ ( ϕ ) = − 3 4 ( f $^{′}$( ϕ ) f ( ϕ ) ) 2 + 1 2 f ′′ ( ϕ ) f ( ϕ ) + ˜ Φ $^{′′}$( ϕ ) ˜ Φ ( ϕ ) (8) R ( ρ ) = ρ $^{υ}$U ( ρ ) and g ( ρ ) = b 2 ρ 2 υ $^{+1}$; υ, b ∈ R , (9) R $^{′′}$( ρ ) R ( ρ ) − ( g $^{′}$( ρ ) g ( ρ ) − 1 ρ ) R $^{′}$( ρ ) R ( ρ ) = U $^{′′}$( ρ ) U ( ρ ) − υ ( υ + 1) ρ 2 . (10) It should be noted here that the choice of g ( ρ ) = bρ 2 υ $^{+1}$/ 2 in (9) is manifestly mandated by the elimination of the first-order derivative of U ( ρ ). 6