  In the present paper we consider an optical system with a $\chi^{(2)}$-type
nonlinearity and unspecified $\mathcal{PT}$-symmetric potential functions.
Considering this as an inverse problem and positing a family of exact solutions
in terms of cnoidal functions, we solve for the resulting potential functions
in a way that ensures the potentials obey the requirements of
$\mathcal{PT}$-symmetry. We then focus on case examples of soliton and periodic
solutions for which we present a stability analysis as a function of their
amplitude parameters. Finally, we numerically explore the nonlinear dynamics of
the associated waveforms to identify the outcome of the relevant dynamical
instabilities of localized and extended states.
