Structure-Preserving Flows of Symplectic Matrix Pairs

We construct a nonlinear differential equation of matrix pairs $(\mathcal{M}(t),\mathcal{L}(t))$ that is invariant (the $\mathbf{\text{Structure-Preserving}}\mathbf{\text{Property}}$) in the class of symplectic matrix pairs \begin{align*} $\mathbb{S}${$\mathcal{S}$1,$\mathcal{S}$2}=\left{\left($\mathcal{M}$,$\mathcal{L}$\right)| \ $\mathcal{M}$=\left[% \begin{array}{cc} X{12} & 0 X{22} & I \end{array}% \right]$\mathcal{S}$2, $\mathcal{L}$=\left[% \begin{array}{cc} I & X{11} 0 & X{21} \end{array}% \right]$\mathcal{S}$1\right.\nonumber \left. $\text{ and }$X=\left[% \begin{array}{cc} X{11} & X_{12} X_{21} & X_{22} \end{array}% \right]$\text{is Hermitian}$\right} \end{align*} for certain fixed symplectic matrices ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$. Its solution also preserves invariant subspaces on the whole orbit (the $\mathbf{\text{Eigenvector-Preserving}}\mathbf{\text{Property}}$). Such a flow is called a $\mathit{\text{structure-preserving}}\mathit{\text{flow}}$ and is governed by a Riccati differential equation (RDE). In addition, Radon’s lemma leads to an explicit form. Therefore, blow-ups for the structure-preserving flows may happen at a finite $t$. To continue, we then utilize the Grassmann manifolds to extend the domain of the structure-preserving flow to the whole $\mathbb{R}$ subtracting some isolated points.