  In [LS], it is shown shown that the first eigenvalue of the Laplacian
restricted to the space of invariant functions on a toric K\"ahler manifold
(i.e. $\lambda_1^\mathbb{T}$, the invariant first eigenvalue) is an unbounded
function of the toric K\"ahler metric. In this note we show that, seen as a
function on the space of toric K\"ahler metrics on a fixed toric manifold,
$\lambda_1^\mathbb{T}$ admits no analytic critical points. We also show that on
$S^2$, the first eigenvalue of the Laplacian restricted to the space of
$S^1$-equivariant functions of any given integer weight admits no critical
points.
