Let $v\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ $(n\geq 2)$ a vector field, such that the set $E=\{v=0\}$ is a manifold of dimension $n-2$. Assume that for every $x\in\mathbb{R}^n-E$, the trajectory $(x_t)_{t\geq 0}$ such that $x_0=x$ and $\dot{x}_t=v(x_t)$ for $t\geq 0$, is periodic, with period $T(x)$, such that $T\in\mathcal{C}^0(\mathbb{R}^n-E,\mathbb{R}_+)$ (is $T$ automatically continuous?). Assume that for every $x\in E$ there exist an antisymmetric matrix $A(x)$ of rank 2 such that $A\in\mathcal{C}^0\big(E, \mathcal{A}_n(\mathbb{R})\big)$ (for the natural topologies of these spaces) and that for a $x'\in\mathbb{R}^n-E$ neighbor of $x$

$$v(x')=A(x)x'+o(x'-x).$$

Question: Does $T$ have a continuous extension defined over $\mathbb{R}^n$?

Let $v\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ $(n\geq 2)$ a vector field, such that the set $E=\{v=0\}$ is a manifold of dimension $n-2$. Assume that for every $x\in\mathbb{R}^n-E$, the trajectory $(x_t)_{t\geq 0}$ such that $x_0=x$ and $\dot{x}_t=v(x_t)$ for $t\geq 0$, is periodic, with period $T(x)$, such that $T\in\mathcal{C}^0(\mathbb{R}^n-E,\mathbb{R}_+)$. Assume that for every $x\in E$ there exist an antisymmetric matrix $A(x)$ of rank 2 such that $A\in\mathcal{C}^0\big(E, \mathcal{A}_n(\mathbb{R})\big)$ (for the natural topologies of these spaces) and that for a $x'\in\mathbb{R}^n-E$ neighbor of $x$

$$v(x')=A(x)x'+o(x'-x).$$

Question: Does $T$ have a continuous extension defined over $\mathbb{R}^n$?

Let $v\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ $(n\geq 2)$ a vector field, such that the set $E=\{v=0\}$ is a manifold of dimension $n-2$. Assume that for every $x\in\mathbb{R}^n-E$, the trajectory $(x_t)_{t\geq 0}$ such that $x_0=x$ and $\dot{x}_t=v(x_t)$ for $t\geq 0$, is periodic, with period $T(x)$, such that $T\in\mathcal{C}^0(\mathbb{R}^n-E,\mathbb{R}_+)$ (is $T$ automatically continuous?). Assume that for every $x\in E$ there exist an antisymmetric matrix $A(x)$ such that $A\in\mathcal{C}^0\big(E, \mathcal{A}_n(\mathbb{R})\big)$ (for the natural topologies of these spaces) and that for a $x'\in\mathbb{R}^n-E$ neighbor of $x$

$$v(x')=A(x)x'+o(x'-x).$$

Question: Does $T$ have a continuous extension defined over $\mathbb{R}^n$?

Let $v\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ $(n\geq 2)$ a vector field, such that the set $E=\{v=0\}$ is a manifold of dimension $n-2$. Assume that for every $x\in\mathbb{R}^n-E$, the trajectory $(x_t)_{t\geq 0}$ such that $x_0=x$ and $\dot{x}_t=v(x_t)$ for $t\geq 0$, is periodic, with period $T(x)$, such that $T\in\mathcal{C}^0(\mathbb{R}^n-E,\mathbb{R}_+)$ (is $T$ automatically continuous?). Assume that for every $x\in E$ there exist an antisymmetric matrix $A(x)$ of rank 2 such that $A\in\mathcal{C}^0\big(E, \mathcal{A}_n(\mathbb{R})\big)$ (for the natural topologies of these spaces) and that for a $x'\in\mathbb{R}^n-E$ neighbor of $x$

$$v(x')=A(x)x'+o(x'-x).$$

Question: Does $T$ have a continuous extension defined over $\mathbb{R}^n$?