(TE)-structures over the 2-dimensional globally nilpotent F-manifold

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We find formal and holomorphic normal forms for a class of meromorphic connections (the so called $(TE)$-structures) over the germ $\mathcal N_{2}$ at the origin of the $2$-dimensional globally nilpotent $F$-manifold. In order to obtain the holomorphic normal forms we prove that the restriction of any $(TE)$-structure $\nabla$ over $\mathcal N_{2}$ at the origin $0\in \mathcal N_{2}$ is either regular singular (in which case $\nabla$ is holomorphically isomorphic to its formal normal forms) or is holomorphically isomorphic to a Malgrange universal connection (in rank two, with pole of Poincar\'{e} rank one). We develop a careful treatment for such Malgrange universal connections. We find normal forms for Euler fields on $\mathcal N_{2}$ and we use them to answer the question when a given Euler field on $\mathcal N_{2}$ is induced by a $(TE)$-structure.

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