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If every regular circle has trivial holonomy, the preceding calculation says that the continuous function $f^{\prime }$ takes values in $\mathbb{Z}$. On a connected radial interval it must therefore be constant. Integrating gives the complete familyConversely, 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.

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.