  We address two mathematical aspects of the Bir\'o--N\'eda dynamical model,
recently applied in the statistical analysis of several and varied complex
phenomena. First, we show that a given implicit assumption ceases to be valid
outside the most simple and common cases, and we analyze the consequences
thereof, in what the formulation of the model and probability conservation is
concerned. Second, we revisit the transient behavior in the case of a constant
reset rate and a constant or linear growth rate, improving on a previous
analysis by including more general initial conditions.
