Flat space limits of symmetric Taub-NUT instantons

We construct $U(1)$-symmetric $SU(2)$ instantons on the Taub-NUT space with generic holonomy, and instanton number and magnetic charge $(k,m)=(2,0)$, $(1,1)$, and $(2,-1)$. We construct these instantons via the bow construction, a Nahm-like non-linear transformation which identifies them with a set of bow data solving an integrable ODE and a series of algebraic relations on an interval. We show how the bow data recovers the data for all corresponding $U(1)$-symmetric instantons on flat $\mathbb{R}^4$ and calorons on $\mathbb{R}^3\times S^1$ in various limits, and we demonstrate the existence of Taub-NUT instantons that do not limit to $\mathbb{R}^4$ instantons nor calorons.