Suppose $z=0$ is a regular singular point of a linear second-order homogeneous ordinary differential equation in the complex plane. Define the monodromy matrix $M$ around $z=0$.

Demonstrate that if

$$M=\left(\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right)$$

then the differential equation admits a solution of the form $a(z)+b(z) \log z$, where $a(z)$ and $b(z)$ are single-valued functions.