On an elliptic system with symmetric potential possessing two global minima

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We consider the system Δu - Wᵤ (u) = 0, for u: R² -> R², W: R² -> R, where Wᵤ (u) is a smooth potential, symmetric with respect to the u₁, u₂ axes, possessing two global minima a±:= (,0) and two connections e±(x₁) connecting the minima. We prove that there exists an equivariant solution u(x₁, x₂) satisfying u(x₁, x₂) -> a±, as x₁ ->, and u(x₁, x₂) -> e±(x₁), as x₂ ->. The problem above was first studied by Alama, Bronsard, and Gui under related hypotheses to the ones introduced in the present paper. At the expense of one extra symmetry assumption, we avoid their considerations with the normalized energy and strengthen their result. We also provide examples for W.