Uniqueness and shooting classification of non-collapsed steady solitons on line bundles

Let $L_k$ be a line bundle over $(\hat{M},\omega)$, with $k>p(\hat{M},\omega)$, and consider the class of metrics specified in the paper, normalized by $b(0)=1$. A steady soliton is non-collapsed if it is the complete non-collapsed steady soliton described in the main theorem. Uniqueness and shooting conjecture. The complete non-collapsed steady soliton is unique up to scaling and isometry in this metric class. Moreover, there exists $f^{\ast}_0\in\mathbb{R}$ such that $f”(0)>f^{\ast}_0$ yields an incomplete metric, $f”(0)=f^{\ast}_0$ yields a complete non-collapsed steady soliton, and $f”(0)<f^{\ast}_0$ yields a complete collapsed steady soliton. The conjecture is supported by numerical integration of the soliton equations, but no proof or resolution is given here.