Consider a system of ordinary differential equations of the form
$$
\dot{x}(t) + \frac{1}{t}Ax(t) = Q(x(t))
$$
where $x(t) \in \mathbb{C}^n$, $A \in \mathrm{Mat}\_{n\times n}(\mathbb{C})$ is a constant matrix, and $Q: \mathbb{C}^n \to \mathbb{C}^n$ is homogeneous of degree $2$, i.e. $Q(\lambda x) = \lambda^2 Q(x)$ for $\lambda \in \mathbb{C}$.


What is known about existence of solutions near $t = 0$?


If it were not for the quadratic term $Q$, the point $t = 0$ would be a **regular singular point** of the ODE and then we could use the Frobenius method. But in all the references I know, regular singular points are only discussed for *linear* systems.



Regular singular point of non-linear ODE: $\dot{x}(t) + t^{-1}Ax(t) = Q(x(t))$
There's nothing inherently linear about constructing power series solutions à la Frobenius. The existence and uniqueness theory for a class of singular non-linear ODEs, of which yours is a special case, is treated for instance in Ch.IX of



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> Wasow, W., *Asymptotic expansions for ordinary differential equations*, (Dover, 1987) reprint from 1965. [ZBL0169.10903](https://zbmath.org/?q=an:0169.10903)
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