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

Some case example exact solutions for quadratically nonlinear optical media with $\mathcal{PT}$-symmetric potentials