An approach to the Ginzburg-Landau problem for superconducting regular polygons is developed making use of an analytical gauge transformation for the vector potential A which gives A(n)=0 for the normal component along the boundary line of different symmetric polygons. As a result the corresponding linearized Ginzburg-Landau equation reduces to an eigenvalue problem in the basis set of functions obeying Neumann boundary condition. Such basis sets are found analytically for several symmetric structures. The proposed approach allows for accurate calculations of the order parameter distributions at low calculational cost (small basis sets) for moderate applied magnetic fields. This is illustrated by considering the nucleation of superconductivity in squares, equilateral triangles and rectangles, where vortex patterns containing antivortices are obtained on the T-c-H phase boundary. The calculated phase boundaries are compared with the experimental T-c(H) curves measured for squares, triangles, disks, rectangles, and loops. The stability of the symmetry consistent solutions against small deviations from the phase boundary line deep into the superconducting state is investigated by considering the full Ginzburg-Landau functional. It is shown that below the nucleation temperature symmetry-switching or symmetry-breaking phase transitions can take place. The symmetry-breaking phase transition has the same structure as the pseudo-Jahn-Teller instability of high symmetry nuclear configurations in molecules. The existence of these transitions is predicted to be strongly dependent on the size of the samples. (c) 2005 American Institute of Physics.