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In this paper, we study the real hypersurfaces $M$ in $\mathbb C^2$ at points$p\in M$ of infinite type. The degeneracy of $M$ at $p$ is assumed to be theleast possible, namely such that the Levi form vanishes to first order in theCR transversal direction. A new phenomenon, compared to known normal forms inother cases, is the presence of resonances as roots of an universal polynomialin the $7$-jet of the defining function of $M$. The main result is a complete(formal) normal form at points $p$ with no resonances. Remarkably, our normalform at such infinite type points resembles closely the Chern-Moser normal format Levi-nondegenerate points. For a fixed hypersurface, its normal forms areparametrized by $S^1\times \mathbb R^*$, and as a corollary we find that theautomorphisms in the stability group of $M$ at $p$ without resonances aredetermined by their $1$-jets at $p$. In the last section, as a contrast, wealso give examples of hypersurfaces with arbitrarily high resonances thatpossess families of distinct automorphisms whose jets agree up to the resonantorder.

A normal form for 1-infinite type hypersurfaces in $\mathbb C^2$. I. Formal Theory