Let $B(-,-): \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$ be a symmetric bilinear map. I am interested in the system of ODEs:


$\frac{dx}{dt} = B(x,x)$


Assume there exists some inner product $g(-,-)$ in $\mathbb{R}^n$ with respect to which $B$ satisfies:


$g(B(x,x),x)=0, \text{ for all $x \in \mathbb{R}^n$.}$


Thinking of $B(x,x)$ as defining a vector field $X$ at $x$, we see that the previous condition implies that $X$ is tangent to unit $g$-sphere $S$ in $\mathbb{R}^n$:


$S = \{x\in \mathbb{R}^n; g(x,x) = 1\}$


My question is this. Is it true that for any such $B$, satisfying the previous hypotheses, integral curves of the restriction of $X$ to $S$ always have finite lengths with respect to $g$, as parametrized curves? Or does there exist some $B$, for which there exists at least one integral curve of the restriction of $X$ having infinite length with respect to $g$?



A question on homogeneous quadratic vector fields
Upon emailing Prof. Yorke, he pointed to me his 2017 article with Saiki and Sander, "Generalized Lorenz equations on a three-sphere", which contains some examples of homogeneous quadratic vector fields, satisfying the condition above (that the vector fields are tangent to the spheres centered at the origin), but which have some trajectories of infinite lengths. This answers my question.


PS: there is no need to give me credit (meaning points) for this answer. The credit all goes to Prof. J. Yorke for this answer.


