Nonlinear equations with superposition formulas and exceptional group G2. III. The superposition formulas

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Abstract
Superposition formulas are derived expressing the general solution of several different systems of nonlinear ordinary differential equations in terms of a fundamental set of particular solutions. The equations, as well as the superposition formulas, are induced by the action of the exceptional Lie group G2 (complex or real) on a homogeneous space G2/G, where G⊆G2 is a maximal subgroup of G2. When G is either parabolic, or simple, three particular solutions are needed. When G is SL(2,C)×SL(2,C) (or one of its real forms), then two particular solutions suffice.