  In this paper, we explore the local geometry of dynamical systems
$\dot{x}=F(x)$ with real time parameterization, where $F$ is holomorphic on
connected open subsets of $\mathbb{C}\stackrel{\sim}{=}\mathbb{R}^2$. We
describe the geometry of first-order equilibria. For equilibria of higher
orders, we establish an equivalent condition for "definite directions",
allowing us to reverse the implication in Theorem 2 of Chapter 2.10 in
[Differential equations and dynamical systems, Lawrence Perko (1990)] under the
additional condition of holomorphy. This enables the geometric construction of
a finite elliptic decomposition. We derive a holomorphic
Poincar\'e-Bendixson-type theorem, leading to the conclusion that bounded
non-periodic orbits are always homoclinic or heteroclinic.
