= Solution
Continuation around one positively oriented loop induces an invertible linear map $M$ on the two-dimensional solution space. Its inverse is continuation around the reversed loop. Over $\mathbb C$, choose an <eigenvector> with <eigenvalue> $\lambda\ne0$, and let $w$ be the corresponding nonzero solution. After continuation it becomes $\lambda w$.
Choose $\sigma$ with $e^{2\pi i\sigma}=\lambda$. The product $h(z)=z^{-\sigma}w(z)$ has trivial <monodromy>, so it is single-valued and <holomorphic> on the punctured disc. Its <Laurent series>, convergent on compact subannuli, gives the <Monodromy eigenfunction Laurent representation>
$$
\boxed{w(z)=z^\sigma\sum_{n=-\infty}^{\infty}c_nz^n.}
$$
No assumption of a <regular singular point> has been made, so the negative part need not terminate.
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