A multi-interval quasi-differential system ${I_{r},M_{r},w_{r}:rinOmega}$ consists of a collection of real intervals, ${I_{r}}$, as indexed by a finite, or possibly infinite index set $Omega$ (where $mathrm{card} (Omega)geqaleph_{0}$ is permissible), on which are assigned ordinary or quasi-differential expressions $M_{r}$ generating unbounded operators in the Hilbert function spaces $L_{r}^{2}equiv L^{2}(I_{r};w_{r})$, where $w_{r}$ are given, non-negative weight functions.For each fixed $rinOmega$ assume that $M_{r}$ is Lagrange symmetric (formally self-adjoint) on $I_{r}$ and hence specifies minimal and maximal closed operators $T_{0,r}$ and $T_{1,r}$, respectively, in $L_{r}^{2}$. However the theory does not require that the corresponding deficiency indices $d_{r}^{-}$ and $d_{r}^{+}$ of $T_{0,r}$ are equal (e. g. the symplectic excess $Ex_{r}=d_{r}^{+}-d_{r}^{-}neq 0$), in which case there will not exist any self-adjoint extensions of $T_{0,r}$ in $L_{r}^{2}$. In this paper a system Hilbert space $mathbf{H}:=sum_{r,in,Omega}oplus L_{r}^{2}$ is defined (even for non-countable $Omega$) with corresponding minimal and maximal system operators $mathbf{T}_{0}$ and $mathbf{T}_{1}$ in $mathbf{H}$.Then the system deficiency indices $mathbf{d}^{pm} =sum_{r,in, Omega}d_{r}^{pm}$ are equal (system symplectic excess $Ex=0$), if and only if there exist self-adjoint extensions $mathbf{T}$ of $mathbf{T}_{0}$ in $mathbf{H}$. The existence is shown of a natural bijective correspondence between the set of all such self-adjoint extensions $mathbf{T}$ of $mathbf{T}_{0}$, and the set of all complete Lagrangian subspaces $mathsf{L}$ of the system boundary complex symplectic space $mathsf{S}=mathbf{D(T}_{1})/mathbf{D(T}_{0})$. This result generalizes the earlier symplectic version of the celebrated GKN-Theorem for single interval systems to multi-interval systems. Examples of such complete Lagrangians, for both finite and infinite dimensional complex symplectic $mathsf{S}$, illuminate new phenoma for the boundary value problems of multi-interval systems. These concepts have applications to many-particle systems of quantum mechanics, and to other physical problems.