  We present an exhaustive class of states with quantum backflow -- the
phenomenon in which a state consisting entirely of positive momenta may have
negative current and the probability flows in the opposite direction to the
momentum. They are characterized by a general function of momenta subject to
very weak conditions. Such a family of states is of interest in the light of a
recent experimental proposal to measure backflow. We find one particularly
simple state which has surprisingly large backflow -- about 41 percent of the
lower bound on flux derived by Bracken and Melloy. We study the eigenstates of
a regularized current operator and we show how some of these states, in a
certain limit, lead to our class of backflow states. This limit also clarifies
the correspondence between the spectrum of the regularized current operator,
which has just two non-zero eigenvalues in our chosen regularization, and the
usual current operator.
