  In this paper we construct (locally) $D$-optimal designs for a wide class of
non-linear multiple regression models, when the design region is a
$k$-dimensional ball. For this construction we make use of the concept of
invariance and equivariance in the context of optimal designs. As examples we
consider Poisson and negative binomial regression as well as proportional
hazard models with censoring. By generalisation we can extend these results to
arbitrary ellipsoids.
