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Circular symmetry in the Hitchin system

To study circularly symmetric field configurations in the SU(2) Hitchin system an SO(2) symmetry, [J_3, ϕ]=0 and [J_3, A_{\pm}]=\pm A_{\pm}, is imposed on the Higgs scalar ϕand the gauge fields A_{\pm} of the system, respectively, where J_3 is a sum of the third components of the orbital angular momenta and the generators of the SU(2). The circular symmetry and the equation \bar{D}ϕ=0 yield onstant, generally nonzero, vacuum expectation values for {\rm Tr}(ϕ^{2}). The equation 4F_{z\bar{z}}=[ϕ, ϕ^{*}] yields a system of differential equations which govern the circularly symmetric field configurations and an exact solution to these equations in a pure gauge form with nontrivial Higgs scalar is obtained.