dimensional Minkowski space which is thus, a soliton of the flow as is expected. For k = 0 we rescale the variables using – with ¯ Ω ≥ 0 and the Eq.(36) becomes – Here, the upper signs are for α ′ > 0 and the lower ones for α ′ < 0, a convention we shall adopt in all that follows. For either case this equation is of the form of Eq.(27) in Prop.(IV.2). Note that now a term of the form ¯ Ω $^{n}$− 1 corresponds to O ( α ′ $^{n}$) in the original flow Eq.(1). We can solve Eq.(38) for all orders and obtain explicit solutions. At N = 1 order, the flow is essentially an un-normalized Ricci flow independent of α ′ with the solution – For N = 2 there exists a fixed point soliton for ¯ Ω = ξ$_{2}$ = 1 and when ¯ Ω = 1 we get – Order N = 3 has no fixed points and the solution is – The ODE at N = 4 has one fixed point at ¯ Ω = ξ$_{4}$ ≈ 1 . 5636. To obtain the solutions for ¯ Ω = ξ$_{4}$ we use the following factorization and partial fraction splittings – 18 ¯ λ + C$_{3}$ = ¯ Ω ± 1 2 ln ∣ ¯ Ω 2 ∓ ¯ Ω + 2 ∣ − 3 √$_{7}$ tan − 1 ( 2 Ω ∓ 1 √$_{7}$ ) (41) ¯ λ + C$_{2}$ = ¯ Ω ± ln ∣ ¯ Ω ∓ 1 ∣ (40) ¯ λ + C$_{1}$ = ¯ Ω (39) 2. Solitons, Solutions and Singularities d ¯ Ω d ¯ λ = 1 ∓ ¯ Ω − 1 + 2 ¯ Ω − 2 ∓ 3 + 5 ζ (3) 2 ¯ Ω − 3 (38) ¯ Ω = Ω | α ′ | k 2 ; ¯ λ = 8 λ | α ′ | (37)