  Using the exact Bethe Ansatz solution, we investigate methods for calculating
the ground-state energy for the $p + ip$-pairing Hamiltonian. We first consider
the Hamiltonian isolated from its environment (closed model) through two forms
of Bethe Ansatz solutions, which generally have complex-valued Bethe roots. A
continuum limit approximation, leading to an integral equation, is applied to
compute the ground-state energy. We discuss the evolution of the root
distribution curve with respect to a range of parameters, and the limitations
of this method. We then consider an alternative approach that transforms the
Bethe Ansatz equations to an equivalent form, but in terms of the real-valued
conserved operator eigenvalues. An integral equation is established for the
transformed solution. This equation is shown to admit an exact solution
associated with the ground state. Next we discuss results for a recently
derived Bethe Ansatz solution of the open model. With the aforementioned
alternative approach based on real-valued roots, combined with mean-field
analysis, we are able to establish an integral equation with an exact solution
that corresponds to the ground-state for this case.
