We consider a type of hypothetical compound materials in which its refractive index in spatial distribution meets $n(-x)=-n^{*}\left(x\right)$, belonging to anti-parity-time- ($\mathrm{A}\mathcal{PT}$) symmetric structures. Additionally, we demand balanced real positive and negative permeabilities with $\mu (-x)=-\mu \left(x\right)$. By introducing parametrization into an $\mathrm{A}\mathcal{PT}$-symmetric transfer matrix, together with a reciprocity theorem, we propose a generic parametric space to display its associated scattering results such as symmetry phase, exceptional point (EP), and symmetry-broken phase. The outcome is irrespective of any system complexity, geometries, materials, and operating frequency. With the parametric space, we find that an $\mathrm{A}\mathcal{PT}$-symmetric system not only enables coherent perfect absorption or lasing occurring at an exceptional point, but also realizes a simultaneous coherent perfect absorption and lasing. Since an $\mathrm{A}\mathcal{PT}$-symmetric system is constructed by balanced positive and negative index materials, the phase accumulated from an optical path length is null, resulting in a real transmission coefficient and an assignment of mode order lost. Although it is challenging to realize negative index materials, a dielectric heterostructure with mirror symmetry is proposed to effectively mimic the electromagnetic responses. Several all-dielectric heterostructures with coherent perfect absorption at an EP, lasing at an EP, and coherent perfect absorption lasing are demonstrated to support our findings on $\mathrm{A}\mathcal{PT}$-symmetric systems.

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• Negative refraction
• Non-Hermitian systems
• PT-symmetry