our opinion fact is that generally speaking $\cT_a\neq\tmi$ (for more details see Proposition \[pr4.11a\] and Example \[ex4.11b\]).

Next assume that $a$ is a regular endpoint for the canonical system , $$\begin{gathered}
\nu_+:=\dim\ker (i J-I)(=\dim H), \qquad \nu_-:=\dim\ker (i J+I)(=\dim (\hat
H\oplus H))\end{gathered}$$ and let $\nu_{b+} $ and $\nu_{b-} $ be indices of inertia of the skew-Hermitian form $[y,z]_b$ (for simplicity assume that $\nu_{b+}\geq \nu_{b-}$). The equality $\Tmi=T_a$ enables us to describe all proper extensions of $\Tmi$ in terms of boundary conditions. For this aim we use a special boundary relation for $\Tma$ which we call decomposing. This boundary relation is defined as follows.

Let $\cH_b$ and $\hat\cH_b$ be finite-dimensional Hilbert spaces and let $$\begin{gathered}
\label{1.15}
\G_b=(\G_{0b}:\,  \hat\G_b:\,  \G_{1b})^\top:\dom\tma\to
\cH_b\oplus\hat\cH_b\oplus \cH_b\end{gathered}$$ be a surjective linear map such that $$\begin{gathered}
\label{1.16}
[y,z]_b=i (\hat\G_b y, \hat\G_b z)-(\G_{1b}y,\G_{0b}z)+(\G_{0b}y,\G_{1b}z),
\quad y,z \in \dom\tma.\end{gathered}$$ (it is not difficult to prove the existence of such a map $\G_b$). Moreover, for each function $y\in\dom\tma$ let $$\begin{gathered}
\label{1.17}
y(t)=\{y_0(t),\, \hat y(t), \, y_1(t)  \}\end{gathered}$$ be the representation of $y(t)$ in accordance with the decomposition . Then the decomposing boundary relation $\G:(\LI)^2\to\HH$ for $\Tma$ is of the form $$\begin{gathered}
\label{1.18}
\G=\lb \lb {\pi y\choose \pi f}, {\G_0' y\choose \G_1'y } 