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In this paper, we first classify all irreducible modules of the vertexalgebra $V_L^+$ when $L$ is a negative definite even lattice of arbitrary rank.In particular, we show that any irreducible $V_L^+$-module is isomorphic to asubmodule of an irreducible twisted $V_L$-module. We then extend this result toa vertex algebra $V_L^+$ when $L$ is a nondegenerate even lattice of finiterank.

Classification of irreducible modules of the vertex algebra $V_L^+$ when $L$ is a nondegenerate even lattice of an arbitrary rank