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A quantum system of a particle interacting with a (non-Abelian) gauge field on the non-simply connected domain M=R^2\{an}Nn=1 is considered, where an, n=1,...,N, are fixed isolated points in R^2. The gauge potential A of the gauge field is a p×p anti-Hermitian matrix-valued 1-form on M, and may be strongly singular at the points an, n=1,...,N. If A is flat, then the physical momentum and the position operators {Pj,qj}2j=1 of the particle satisfy the canonical commutation relations (CCR) with two degrees of freedom on a suitable dense domain of the Hilbert space L^2(R^2;C^p). A necessary and sufficient condition for this representation to be the Schrödinger 2-system is given in terms of the Wilson loops of the rectangles not intersecting an, n=1,...,N. This also gives a characterization for the representation to be non-Schrödinger. It is proven that, for a class of gauge potentials, which is not necessarily flat, Pj is essentially self-adjoint. Moreover, an example, which gives a class of non-Schrödinger representations of the CCR with two degrees of freedom, is discussed in some detail

Gauge theory on a non-simply connected domain and representations of canonical commutation relations