  We study the classification of 2-dimensional scale-invariant rigid special
Kahler (RSK) geometries, which potentially describe the Coulomb branches of N=2
supersymmetric field theories in four dimensions. We show that this
classification is equivalent to the solution of a set of polynomial equations
by using an integrability condition for the central charge, scale invariance,
constraints coming from demanding single-valuedness of physical quantities on
the Coulomb branch, and properties of massless BPS states at singularities. We
find solutions corresponding to lagrangian scale invariant theories--including
the scale invariant G_2 theory not found before in the literature--as well as
many new isolated solutions (having no marginal deformations). All our
scale-invariant RSK geometries are consistent with an interpretation as
effective theories of N=2 superconformal field theories, and, where we can
check, turn out to exist as quantum field theories.
