Galois lines for a canonical curve of genus 4, II: Skew cyclic lines

Let $C \subset \mathbb{P}^3$ be a canonical curve of genus $4$ over an algebraically closed field $k$ of characteristic zero. For a line $l$, we consider the projection $\pi\_l\colon C \rightarrow \mathbb{P}^1$ with center $l$ and the extension of the function fields $\pi^\_l\colon k(\mathbb{P}^1) \hookrightarrow k(C)$. A line $l$ is referred to as a cyclic line if the extension $k(C)/\pi\_l^(k(\mathbb{P}^1))$ is cyclic. A line $l \subset \mathbb{P}^3$ is said to be skew if $C \cap l = \emptyset$. We prove that the number of skew cyclic lines is equal to $0,1,3$ or $9$. We determine curves that have nine skew cyclic lines.