  We study non-interacting fermionic systems dissipatively driven at their
boundaries, focusing in particular on the case of a non-number-conserving
Hamiltonian, which for example describes an $XY$ spin chain. We show that
despite the lack of number conservation, it is possible to convert the problem
of calculating the normal modes of the master equations and their corresponding
rapidities, into diagonalizing simply an $L\times L$ tridiagonal bordered
$2-$Toeplitz matrix, where $L$ is the size of the system. Such structure of
matrix allows us to further reduce the problem into solving a scalar
trigonometric non-linear equation for which we also show, in the case of an
Ising chain, exact analytical explicit, and system size independent, solutions.
