First double mode of the Poisson distribution of order $k$

The Poisson distribution of order $k$ is a special case of a compound Poisson distribution. For $k=1$ it is the standard Poisson distribution. Our focus in this note is for $k\ge2$. For sufficiently small values of the rate parameter $\lambda$, both the median and mode equal zero. The median is zero if and only if $\lambda \le (\ln2)/k$. The supremum value of $\lambda$ for the mode to be zero is known only for small values of $k$. This note presents results for the "first double mode" by which is meant the first occasion (smallest value of $\lambda$) the distribution is bimodal, with modes at $0$ and $m>0$. Next, an almost complete answer is supplied to the question "which positive integers cannot be modes of the Poisson distribution of order $k$?" The term "almost complete" signifies that some parts of the answer are conjectures based on numerical searches. However, if the conjectures are proved to be correct, the solution presented in this note is complete: all parameter values are covered.