Since a general point in phase - space is the restriction to the hypersurface τ = 0 of the general solution we can identify the space S C of ( complexified ) solutions with ( complexified ) phasespace P ∞ C . Furthermore , the subset S V C ⊂ S C of solutions satisfying Virasoro and static gauge conditions can be identified with the second class constraint surface P V C ⊂ P ∞ C defined by these conditions . We can describe the map ( 4.1 ) as an embedding M ( 2g+2 ) C ֒→ P V C . If we further impose reality conditions by restricting the algebro - geometric data M ( 2g+2 ) C to real algebro - geometric data ( see [ 13 ] for a detailed discussion of reality conditions ) then finite - gap integration describes an injective map [ 35 ]