content_image/350261.jpg content_image/350260.jpg content_image/350258.jpg content_image/350257.jpg content_image/350256.jpg
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 – 

<img src='content_image/350261.jpg'>

<img src='content_image/350260.jpg'>

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). 

## 2. Solitons, Solutions and Singularities 

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 – 

<img src='content_image/350258.jpg'>

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 – 

<img src='content_image/350257.jpg'>

<img src='content_image/350256.jpg'>

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 – 

