\l\in\bC_\pm.\end{gathered}$$ As was shown in [@LesMal03] the relations $$\begin{gathered}
N_+=n_+(\Tmi)+k_\cN, \qquad N_-=n_-(\Tmi)+k_\cN\end{gathered}$$ hold with $k_\cN=\dim\cN$. In the present paper by using just a fact of existence of a decomposing boundary relation we prove the equalities $$\begin{gathered}
\label{1.20}
N_+=\nu_+ + \nu_{b+}, \qquad N_-=\nu_- + \nu_{b-}.\end{gathered}$$ Formula yields the known estimates $\nu_\pm\leq N_\pm\leq \dim\bH$ obtained by analytic methods in[@Atk; @KogRof75]. Observe also that in a somewhat different way the equalities were proved for definite systems in [@BHSW10 Lemma 4.15].

Existence of the nontrivial multivalued part $\mul\G$ of the decomposing boundary relation is caused by the nontrivial null manifold $\cN$, which can be seen from the equalities $$\begin{gathered}
\label{1.21a}
\mul\G=\{\{\G_0'y,\G_1'y\}:\, y\in\cN\}, \qquad \dim (\mul\G)=k_\cN(=\dim\cN).\end{gathered}$$ Formula implies that for the definite canonical system the decomposing boundary relation turns into the decomposing boundary triplet $\Pi=\bta$ for $\Tma$. In the case $n_+(\Tmi)=n_-(\Tmi)$ this triplet is of the form $\Pi=\bt$ with the boundary Hilbert space $\cH=H\oplus\hat H \oplus \cH_b$ and the operators $\G_j$ given by $$\begin{gathered}
\G_0 \{\wt y, \wt f\} =\{ y_0(a),\, \tfrac i {\sqrt 2} (\hat y(a)-\hat\G_b
y),\,\G_{0b}y \} (\in H\oplus\hat H \oplus \cH_b),\qquad\qquad\qquad\quad
\label{1.22}\\
\G_1 \{\wt y, \wt f\} = \{ y_1(a),\, \tfrac 1 {\sqrt 2} (\hat y(a)+\hat\G_b
y),\,-\G_{1b}y \}(\in H\oplus\hat H \oplus \cH_b), 