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We investigate in the paper general (not necessarily definite) canonicalsystems of differential equation in the framework of extension theory ofsymmetric linear relations. For this aim we first introduce the new notion of aboundary 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 spaceand $\cH_1$ is a subspace in $\cH_0$. Unlike known concept of a boundaryrelation (boundary triplet) for $A^*$ our definition of $\G$ is applicable torelations $A$ with possibly unequal deficiency indices $n_\pm(A)$. Next wedevelop the known results on minimal and maximal relations induced by thegeneral 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 (socalled decomposing) boundary relation for $\Tma$ we describe in terms ofboundary conditions proper extensions of $\Tmi$ in the case of the regularendpoint $a$ 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 atriplet we show that self-adjoint decomposing boundary conditions exist onlyfor Hamiltonian systems; moreover, we describe all such conditions in thecompact form. These results are generalizations of the known results byRofe-Beketov on regular differential operators. We characterize also allmaximal dissipative and accumulative separated boundary conditions, which existfor arbitrary (not necessarily Hamiltonian) definite canonical systems.

Boundary relations and boundary conditions for general (not necessarily definite) canonical systems with possibly unequal deficiency indices