  We show that a two-level non-Hermitian Hamiltonian with constant off-diagonal
exchange elements can be analyzed exactly when the underlying exceptional point
is perfectly encircled in the complex plane. The state evolution of this system
is explicitly obtained in terms of an ensuing transfer matrix, even for large
encirclements, regardless of adiabatic conditions. Our results clearly explain
the direction-dependent nature of this process and why in the adiabatic limit
its outcome is dominated by a specific eigenstate irrespective of initial
conditions. Moreover, numerical simulations suggest that this mechanism can
still persist in the presence of nonlinear effects. We further show that this
robust process can be harnessed to realize an optical omni-polarizer: a
configuration that generates a desired polarization output regardless of the
input polarization state, while from the opposite direction it always produces
the counterpart eigenstate.
