= Solution
If every regular circle has trivial <holonomy>, the preceding calculation says that the continuous function $f'$ takes values in $\mathbb Z$. On a connected radial interval it must therefore be constant. Integrating gives the complete family
$$
\boxed{f(r)=nr+b,\qquad n\in\mathbb Z,\quad b\in\mathbb R,\quad nr+b>0\text{ on the chosen interval}.}
$$
Conversely, every such function has identity <holonomy> around every circle. With the opposite sign convention for $f$, replace the positivity condition by nonvanishing; the <Riemannian metric> is unchanged by changing its sign.
\b[The slope parameter is discrete.] Thus the printed parameter count can mean one integer and one real parameter, but it cannot mean two freely varying real parameters. Allowing $f=ar+b$ with arbitrary real $a$ gives a locally flat family, not trivial circular <holonomy>: for example $a=1/2$ gives a rotation by $\pi$. This distinction is captured by <flat circular metrics with trivial holonomy>.
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