Monodromic singularities without curves of zero angular speed

We consider planar analytic vector fields $\mathcal{X}$ having a monodromic singular point with Poincaré map $\mathrm{\Pi }$. We use the fact that there always exists a real analytic invariant curve $F=0$ of $\mathcal{X}$ in a neighborhood of that singularity. We find some relations between $\mathrm{\Pi }$ and $F$ that can be used to determine new conditions that guarantee the analyticity of $\mathrm{\Pi }$ at the singularity. In the special case that $F$ becomes an inverse integrating factor of $\mathcal{X}$, we rediscover formulas obtained previously by other methods. Applications to the center-focus problem and also to vector fields with degenerate infinity are given.