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abstract: |
    We investigate in the paper general (not necessarily definite) canonical systems of differential equation in the framework of extension theory of symmetric linear relations. For this aim we first introduce the new notion of a boundary relation $\G:\gH^2\to\HH$ for $A^*$, where $\gH$ is a Hilbert space, $A$ is a symmetric linear relation in $\gH, \cH_0$ is a boundary Hilbert space and $\cH_1$ is a subspace in $\cH_0$. Unlike known concept of a boundary relation (boundary triplet) for $A^*$ our definition of $\G$ is applicable to relations $A$ with possibly unequal deficiency indices $n_\pm(A)$. Next we develop the known results on minimal and maximal relations induced by the general canonical system $ J y'(t)-B(t)y(t)=\D (t)f(t)$ on an interval $\cI=(a,b),\; -\infty\leq a<b\leq\infty $ and then by using a special (so called decomposing) boundary relation for $\Tma$ we describe in terms of boundary conditions proper extensions of $\Tmi$ in the case of the regular endpoint $a$ and arbitrary (possibly unequal) deficiency indices $n_\pm
    (\Tmi)$. If the system is definite, then decomposing boundary relation $\G$ turns into the decomposing boundary triplet $\Pi=\bt$ for $\Tma$. Using such a triplet we show that self-adjoint decomposing boundary conditions exist only for Hamiltonian systems; moreover, 