Conical intersections of the 2 E-2＇ and 1 (2)A(1)＇ states of Li-3 are considered as a first step in the description of the excited state involved in the absorption at 14 583 cm(-1). Quite novel results are obtained. In addition to the standard symmetry-required D-3h seam of conical intersection, C-2v symmetry-allowed seams are found and their locus determined as a function the symmetric stretch coordinate. The C-2v seams of conical intersection intersect the D-3h conical intersection seam at two distinct points producing two heptafurcations of the D-3h seam. The locus of the C-2v conical intersection points is explained in terms of a quadratic Jahn-Teller model. The overall geometric phase effect associated with the D-3h and C-2v conical intersections is well described using a previously suggested phase factor deduced from molecular properties. In addition to these conical intersections an intersection of the 1 (2)A(1)＇ and 2( 2)E＇ is states is located.