each $t\in{{\mathbb R}}$: $$\begin{aligned}
  P^{\, t} v := \Pi_{X_t(x)}DX_t v ,
  \quad
  v\in T_x M, t\in{{\mathbb R}}, X(x)\neq 0,\end{aligned}$$ where $\Pi_{X_t(x)}:T_{X_t(x)}M\to N_{X_t(x)}$ is the projection on $N_{X_t(x)}$ parallel to $X(X_t(x))$. We remark that the definition of $\Pi_x$ depends on $X(x)$ and ${\EuScript{J}}_X$ only. The linear Poincaré flow $P^{\,t}$ is a linear multiplicative cocycle over $X_t$ on the set $U$ with the exclusion of the singularities of $X$.

In this setting we can say that the linear Poincaré flow is (strictly) ${\EuScript{J}}$-separated and (strictly) ${\EuScript{J}}$-monotonous using the non-degenerate bilinear form ${\EuScript{J}}$ restricted to $N_x$ for a regular $x\in U$. More precisely: $P^t$ is ${\EuScript{J}}$-monotonous if $\partial_t{\EuScript{J}}(P^tv)\mid_{t=0}\ge0$, for each $x\in U, v\in T_xM\setminus\{0\}$ and $t>0$, and strictly ${\EuScript{J}}$-monotonous if $\partial_t{\EuScript{J}}(P^tv)\mid_{t=0}>0$, for all $v\in T_xM\setminus\{0\}$, $t>0$ and $x\in U$.

\[pr:J-separated-spectrum\] Let $L:V\to V$ be a ${\EuScript{J}}$-separated linear operator. Then

1.  $L$ can be uniquely represented by $L=RU$, where $U$ is a ${\EuScript{J}}$-isometry and $R$ is ${\EuScript{J}}$-symmetric (or ${\EuScript{J}}$-pseudo-adjoint; see Proposition \[pr:propbilinear\]) with positive spectrum.

2.  the operator $R$ can be diagonalized by a ${\EuScript{J}}$-isometry. Moreover the eigenvalues of $R$ satisfy $$\begin{aligned}
          0<r_-^q\le\dots\le r_-^1=r_-\le r_+=r_1^+\le\dots\le r_+^p.
        \end{aligned}$$

3.  the operator $L$ is (strictly) ${\EuScript{J}}$-monotonous if, and only if, $r_-\le (<) 1$ and $r_+\ge (>) 