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Rational solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV)equations of associativity are given in terms a configurations of vectors whichsatisfy certain algebraic conditions known as $\bigvee$-conditions. Thesimplest examples of such configuration are the root systems of finite Coxetergroups. In this paper conditions are derived which ensure that an extendedconfiguration - a configuration in a space one-dimension higher -satisfy these$\bigvee$-conditions. Such a construction utilizes the notion of a small-orbit,as defined by Serganova. Symmetries of such resulting solutions to theWDVV-equations are studied; in particular, Legendre transformations. It isshown that these Legendre transformations map extended-rational solutions totrigonometric solutions and, for certain values of the free data, one obtains atransformation from extended $\bigvee$-systems to the trigonometric almost dualsolutions corresponding to the classical extended affine Weyl groups.

Extended V-systems and almost-duality for extended affine Weyl orbit spaces