  The Complexity of the Thouless-Anderson-Palmer (TAP) solutions of the Ising
$p$-spin is investigated in the temperature regime where the equilibrium phase
is one step Replica Symmetry
  Breaking. Two solutions of the resulting saddle point equations are found.
One is supersymmetric (SUSY) and includes the equilibrium value of the free
energy while the other is non-SUSY. The two solutions cross exactly at a value
of the free energy where the replicon eigenvalue is zero; at low free energy
the complexity is described by the SUSY solution while at high free energy it
is described by the non-SUSY solution. In particular the non-SUSY solution
describes the total number of solutions, like in the
  Sherrington-Kirkpatrick (SK) model. The relevant TAP solutions corresponding
to the non-SUSY solution share the same feature of the corresponding solutions
in the SK model, in particular their Hessian has a vanishing isolated
eigenvalue. The TAP solutions corresponding to the SUSY solution, instead, are
well separated minima.
