  We discuss spectral properties of the equatorial Podles sphere. As a
preparation we also study the `degenerate' (i.e. $q=0$) case (related to the
quantum disk). We consider two different spectral triples: one related to the
Fock representation of the Toeplitz algebra and the isopectral one. After the
identification of the smooth pre-$C^*$-algebra we compute the dimension
spectrum and residues. We check the nontriviality of the (noncommutative) Chern
character of the associated Fredholm modules by computing the pairing with the
fundamental projector of the $C^*$-algebra (the nontrivial generator of the
$K_0$-group) as well as the pairing with the $q$-analogue of the Bott
projector. Finally, we show that the local index formula is trivially
satisfied.
