  In this paper we analyse non-supersymmetric single centred extremal black
hole solutions in N=2 supergravity theory coupled to $n$ vector multiplets with
purely cubic pre-potential in four dimensions. We consider the algebraic
attractor equations in their most general form at the black hole horizon. We
explicitly construct a new class of solutions for these attractor equations.
These solutions are characterised by a set of involutory matrices. These
involutions are obtained from a constraint involving the parameters in the
pre-potential and generate new attractor points in the moduli space.
