  $\mathcal{PT}$ symmetry is a unique platform for light manipulation and
versatile use in unidirectional invisibility, lasing, sensing, etc. Broken and
unbroken $\mathcal{PT}$-symmetric states in non-Hermitian open systems are
described by scattering matrices. A multilayer structure, as a simplest example
of the open system, has no certain definition of the scattering matrix, since
the output ports can be permuted. The uncertainty in definition of the
exceptional points bordering $\mathcal{PT}$-symmetric and
$\mathcal{PT}$-symmetry-broken states poses an important problem, because the
exceptional points are indispensable in applications as sensing and mode
discrimination. Here we derive the proper scattering matrix from the
unambiguous relation between the $\mathcal{PT}$-symmetric Hamiltonian and
scattering matrix. We reveal that the exceptional points of the scattering
matrix with permuted output ports are not related to the $\mathcal{PT}$
symmetry breaking. Nevertheless, they can be employed for finding a lasing
onset as demonstrated in our time-domain calculations and scattering-matrix
pole analysis. Our results are important for various applications of the
non-Hermitian systems including encircling exceptional points, coherent perfect
absorption, $\mathcal{PT}$-symmetric plasmonics, etc.
