  This paper is a continuation of the paper F. A. Arias and M. Malakhaltsev "A
generalization of the Gauss-Bonnet and Hopf-Poincar\'e theorems",
ArXiv:1510.01395 [MathDG] 5 Oct 2015. Let $\pi : E \to M$ be a locally trivial
fiber bundle over a two-dimensional manifold $M$, and $\Sigma \subset M$ be a
discrete subset. A subset $Q \subset E$ is called an $n$-sheeted branched
section of the bundle $\pi$ if $Q' = \pi^{-1}(M \setminus \Sigma) \cap Q$ is a
$n$-sheeted covering of $M \setminus \Sigma$. The set $\Sigma$ is called the
singularity set of the branched section $Q$. We define the index of a
singularity point of a branched section, and give examples of its calculation,
in particular for branched sections of the projective tangent bundle of $M$
determined by binary differential equations. Also we define a resolution of
singularities of a branched section, and prove an analog of
Hopf-Poincar\'e-Gauss-Bonnet theorem for the branched sections admitting a
resolution.
