I need to find the flow of the following vector field in $\mathbb{R}^n$ : $ \mathbf{y} \mapsto \mathbf{y}\cdot\mathbf{n}\mathrm{A}\cdot \mathbf{y}$ where $\mathbf{n}$ is a given vector and $\mathrm{A}$ is a given $n\times n$ invertible matrix.
What I've found so far :
If the dimension $n=1$, then $f_t(x) = x / (1-t*a*x)$ satisfies $\frac{d}{d t} f_t(x) = a*f_t(x)^2$ and $f_0(x)=x$, which answers the question.
In arbitrary dimension $n$, this solution extends to $f_t(\mathbf{x}) = \mathbf{x} / (1-t*\mathbf{n}\cdot\mathbf{x})$, which is the flow of $\mathbf{y} \mapsto \mathbf{n}\cdot\mathbf{y} \mathbf{y}$, which answers the question for the identity matrix.
However, I couldn't further generalize this solution to an arbitrary invertible matrix $\mathrm{A}$.
Any ideas ?