  We build a one-parameter family of S^{1}-invariant metrics on the unit disc
with fixed total area for which the second eigenvalue of the Laplace operator
in the case of both Neumann and Dirichlet boundary conditions is simple and has
an eigenfunction with a closed nodal line. In the case of Neumann boundary
conditions, we also prove that this eigenfunction attains its maximum at an
interior point, and thus provide a counterexample to the hot spots conjecture
on a simply connected surface. This is a consequence of the stronger result
that within this family of metrics any given (finite) number of S^{1}-invariant
eigenvalues can be made to be arbitrarily small, while the non-invariant
spectrum becomes arbitrarily large.
