particular case of the system .

Denote by $\lI$ the semi-Hilbert space of $\bH$-valued Borel functions $f(t)$ on $\cI$ with $\int_\cI (\D(t)f(t),f(t))\, dt < \infty $ and let $(f,g)_\D$ be the semi-definite inner product in $\lI$. Assume also that $\LI$ is the corresponding Hilbert space of equivalence classes and $\pi$ is the quotient map from $\lI$ onto $\LI$, so that the inner product in $\LI$ is $(\wt f, \wt g
)(=(\pi f,\pi g))=(f,g)_\D, \;\; \wt f,\wt g\in \LI$.

The null manifold $\cN$ of the system plays an essential role in our considerations. Recall [@KogRof75] that $\cN$ is defined as the set of all solutions of the equation $J y'(t)-B(t)y(t)=0$ such that $\D(t)y(t)=0$ a.e. on $\cI$. The system is said to be definite if $\cN=\{0\}$ and indefinite in the opposite case.

As is known the extension theory of symmetric relations is the natural framework for boundary value problems involving canonical systems of differential equations (see [@Orc; @LT82; @DLS88; @DLS93; @HSW00; @BHSW10; @LesMal03] and references therein). This framework is based on the concept of minimal and maximal relations which are defined as follows. Let $\tma$ be the set of all pairs $\{y,f\}\in\lI\times \lI $ satisfying the system and let $\cT_0$ be the 