Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank $1$

Holomorphic vector bundles on $\mathbb{C}\times M$, $M$ a complex manifold, with meromorphic connections with poles of Poincaré rank $1$ along $\{0\}\times M$ arise naturally in algebraic geometry. They are called $(TE)$-structures here. This paper takes an abstract point of view. It gives a complete classification of all $(TE)$-structures of rank $2$ over germs $\big(M,t^0\big)$ of manifolds. In the case of $M$ a point, they separate into four types. Those of three types have universal unfoldings, those of the fourth type (the logarithmic type) not. The classification of unfoldings of $(TE)$-structures of the fourth type is rich and interesting. The paper finds and lists also all $(TE)$-structures which are basic in the following sense: Together they induce all rank $2$ $(TE)$-structures, and each of them is not induced by any other $(TE)$-structure in the list. Their base spaces $M$ turn out to be $2$-dimensional $F$-manifolds with Euler fields. The paper gives also for each such $F$-manifold a classification of all rank $2$ $(TE)$-structures over it. Also this classification is surprisingly rich. The backbone of the paper are normal forms. Though also the monodromy and the geometry of the induced Higgs fields and of the bases spaces are important and are considered.