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A 2D Fully Frustrated XY(FFXY) class of models is shown to contain a newgroundstate in addition to the checkerboard groundstates of the standard 2DFFXY model. The spin configuration of this additional groundstate is obtained.Associated with this groundstate there are additional phase transitions. Anorder parameter accounting for these new transitions is proposed. Thetransitions associated with the new order parameter are suggested to be similarto a 2D liquid-gas transition which implies Z_2-Ising like transitions. Thissuggests that the class of 2D FFXY models belongs within a U(1) x Z_2 xZ_2-designation of possible transitions, which implies that there are sevendifferent possible single and combined transitions. MC-simulations for thegeneralized fully frustrated XY (GFFXY) model on a square lattice are used toinvestigate which of these possibilities can be realized in practice: five ofthe seven are encountered. Four critical points are deduced from theMC-simulations, three consistent with central charge c=3/2 and one with c=1.The implications for the standard 2D FFXY-model are discussed in particularwith respect to the long standing controversy concerning the characteristics ofits phase transitions.

Symmetry-allowed phase transitions realized by the two-dimensional fully frustrated XY class