  Master equations describing open quantum dynamics are typically first order
differential equations. When such dynamics brings the trajectories in state
space of more than one initial state to the same point at finite instants in
time, the generator of the corresponding master equation becomes singular. The
first-order, time-local, homogeneous master equations then fail to describe the
dynamics beyond the singular point. Retaining time-locality in the master
equation necessitates a reformulation in terms of higher-order differential
equations. We formulate a method to eliminate the divergent behavior of the
generator by using a combination of higher-order derivatives of the generator
with suitable weights and illustrate it with several examples. We also present
a detailed study of the central spin model and we propose the average rate of
information inflow in non-Markovian processes as a quantity that captures a
different aspect of non-Markovian dynamics.
