was a non-positive vector.

This means that a flow $X_t$ is strictly ${\EuScript{J}}$-separated if, and only if, its time reversal $X_{-t}$ is strictly $(-{\EuScript{J}})$-separated.

A vector field $X$ is ${\EuScript{J}}$-*non-negative* on $U$ if ${\EuScript{J}}(X(x))\ge0$ for all $x\in U$, and ${\EuScript{J}}$-*non-positive* on $U$ if ${\EuScript{J}}(X(x))\leq 0$ for all $x\in U$. When the quadratic form used in the context is clear, we will simply say that $X$ is non-negative or non-positive.

We apply this notion to the linear Poincaré flow defined on regular orbits of $X_t$ as follows.

Suppose that the vector field $X$ is non-negative on $U$. Then, the span $E^X_x$ of $X(x)\neq 0$ is a ${\EuScript{J}}$-non-degenerate subspace.

According to item (1) of Proposition \[pr:propbilinear\], we have that $T_xM=E_x^X\oplus N_x$, where $N_x$ is the pseudo-orthogonal complement of $E^X_x$ with respect to the bilinear form ${\EuScript{J}}$, and $N_x$ is also non-degenerate. Moreover, by the definition, the index of ${\EuScript{J}}$ restricted to $N_x$ is the same as the index of ${\EuScript{J}}$. Thus, we can define on $N_x$ the positive and negative cones with core $N_x^+$ and $N_x^-$, respectively.

Define the Linear Poincaré Flow $P^{\, t}$ of $X_t$ along the orbit of $x$, by projecting $DX_t$ orthogonally (with respect to ${\EuScript{J}}$) over $N_{X_t(x)}$ for 