For Ginzburg-Landau vortices of degree 2 and 3, the only bounded zero modes are the three geometric symmetries, and the Morse indices are 2 and 6.
Beaulieu, Bounded solutions for an ordinary differential system from the Ginzburg–Landau theory,Proc
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Nondegeneracy and Morse Index of Ginzburg--Landau Vortices
For Ginzburg-Landau vortices of degree 2 and 3, the only bounded zero modes are the three geometric symmetries, and the Morse indices are 2 and 6.