Consider the nonlinear map $F\_i:\mathbb R^2 \to \mathbb R$


$F\_i(x):=\varepsilon^2\langle x, A\_i x\rangle +\varepsilon\langle b\_i,x \rangle + x\_i,$


where $A\_i$ is some matrix and $b\_i$ some vector


Can we uniquely solve the equation $(F\_1(x),F\_2(x))=z\_0 \in \mathbb R^2$ for $\varepsilon>0$ small enough such that $x = \sum\_{n=0}^{\infty} \varepsilon^n z\_n$ where $z\_n$ are some coefficients?



Analytic solution of low-dimensional Riccati equation
This is a particular instance of the analytic implicit function theorem; see e.g. [this MO page](https://mathoverflow.net/questions/73388/analytic-implicit-function-theorem) and further references there.


