Given a holomorphic vector bundle $E \to \mathbb{C}^*$, and some linear map $\Gamma: E \to E$, we can define the space $\Sigma := \{ (s,t) : \det \bigg( \Gamma(s) - t \bigg) = 0 \}$, the spectral curve. For each point $z \in \Sigma$, we can attach a fiber, consisting of the eigenlines of $\Gamma$. Generically we have 1 dimensional spaces, so we try to turn this into a line bundle $L \to \Sigma$.
Bad things happen at the branch points of $\Sigma$. The rank of our bundle may change. My question is if anyone has references that analyze this problem?