  The Neumann-Poincar\'e operator is a boundary-integral operator associated
with harmonic layer potentials. This article proves the existence of
eigenvalues within the essential spectrum for the Neumann-Poincar\'e operator
for certain Lipschitz curves in the plane with reflectional symmetry, when
considered in the functional space in which it is self-adjoint. The proof
combines the compactness of the Neumann-Poincar\'e operator for curves of class
$C^{2,\alpha}$ with the essential spectrum generated by a corner. Eigenvalues
corresponding to even (odd) eigenfunctions are proved to lie within the
essential spectrum of the odd (even) component of the operator when a
$C^{2,\alpha}$ curve is perturbed by inserting a small corner.
