These notes are a slightly enlarged version of my habilitation thesis, where our research interest and main results in the past few years are summarized. Most of the discussion revolves around complex ordinary differential equations and their underling foliations, singularity theory and dynamical systems. Compared to the original text, a section containing some background material on holomorphic foliations was added. Also some new results obtained in the past three years that are in line with the one presented in the habilitation were included.

• 32M25 - Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions
• 32S45 - Modifications; resolution of singularities (complex-analytic aspects)
• 32S65 - Singularities of holomorphic vector fields and foliations
• 37F75 - Dynamical aspects of holomorphic foliations and vector fields