homework and exercises - Metric for infinite straight cosmic string

A string theory question on my general relativity problem set:

Metric is given as

$$\mathrm{d}s^2 = -A(r)\mathrm{d}t^2 + B(r)\mathrm{d}r^2 + r^2 \mathrm{d}\theta^2.$$

a) Solve the vacuum equations and find the metric around the massive source assuming that the Ricci curvature vanishes. Use a change of coordinates to put the metric in the form,

$$\mathrm{d}s^2 = -\mathrm{d}t^2 + \mathrm{d}r^2 + C(r)r^2 \mathrm{d} \theta^2$$

b) Consider the metric on a spatial slice ($t$ = constant). Use the relationship between curvature and parallel transport to determine the total curvature that must be contained in the region containing the source if parallel transport around a single cosmic string rotates a vector by $2\pi/N$.

c) Write the metric at a large distance from a configuration of $k$ sources, assuming each separately gives rise to the metric described in the previous part. What is the maximum number of parallel such cosmic strings that can inhabit space-time? With this maximum number of cosmic strings, what is the total curvature over a 2D surface perpendicular to the strings?

--I attempted the first part, found all connections and Ricci tensor components. Requiring that the Ricci tensor be zero. I attempted to get what seemed to be a mess of algebra to solve for $B(r)$ and $A(r)$. But I didn't really get it to work. Regardless, I know that in the form C(r) is given by a few books to be something of the form $1-4\mu$, whose interpretation as canonical spacetime makes enough sense to me. The latter two parts are where I'm really at a loss.