  It is shown that if the decoherence matrix corresponding to a qubit master
equation has a block-diagonal real part, then the evolution is determined by a
one-dimensional oscillator equation. Further, when the full decoherence matrix
is block-diagonal, then the necessary and sufficient conditions for completely
positive evolution may be formulated in terms of the oscillator Hamiltonian or
Lagrangian. When the solution of the oscillator equation is not known, an
explicit sufficient condition for complete positivity can still be obtained,
based on a Hamiltonian/Lagrangian inequality. A rotational form-invariance
property is used to characterise the evolution via a single first-order
nonlinear differential equation, enabling some further exact results to be
obtained. A class of master equations is identified for which complete
positivity reduces to the simpler condition of positivity.
