  Let $V_{L}$ be the vertex algebra associated to a non-degenerate even lattice
$L$, $\theta$ the automorphism of $V_{L}$ induced from the $-1$-isometry of
$L$, and $V_{L}^{+}$ the fixed point subalgebra of $V_{L}$ under the action of
$\theta$. In this series of papers, we classify the irreducible weak
$V_{L}^{+}$-modules and show that any irreducible weak $V_{L}^{+}$-module is
isomorphic to a weak submodule of some irreducible weak $V_{L}$-module or to a
submodule of some irreducible $\theta$-twisted $V_{L}$-module.
  In this paper (Part 1), we show that when the rank of $L$ is $1$, every
non-zero weak $V_{L}^{+}$-module contains a non-zero $M(1)^{+}$-module, where
$M(1)^{+}$ is the fixed point subalgebra of the Heisenberg vertex operator
algebra $M(1)$ under the action of $\theta$.
