Consider a system of ordinary differential equations of the form $$ \dot{x}(t) + \frac{1}{t}Ax(t) = B(x(t)) $$ where $x(t) \in \mathbb{R}^n$, $A \in \mathrm{Mat}_{n\times n}(\mathbb{R})$ is a constant matrix, and $B: \mathbb{R}^n \to \mathbb{R}^n$ is homogeneous of degree $2$, i.e. $B(\lambda x) = \lambda^2 B(x)$ for $\lambda \in \mathbb{R}$.
What is known about existence of solutions near $t = 0$?
If it were not for the quadratic term, 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 books I have, regular singular points are only discussed for linear systems.