  We characterize $D$-optimal designs in the two-dimensional Poisson regression
model with synergetic interaction and provide an explicit proof. The proof is
based on the idea of reparameterization of the design region in terms of
contours of constant intensity. This approach leads to a substantial reduction
of complexity as properties of the sensitivity can be treated along and across
the contours separately. Furthermore, some extensions of this result to higher
dimensions are presented.
