  The crystalline electric field (CEF) is investigated in
Pr$_3$Pt$_{23}$Si$_{11}$ and Nd$_3$Pt$_{23}$Si$_{11}$ by neutron spectroscopy
(NS). At low temperature, the number of observed CEF excitations is consistent
with the orthorhombic symmetry at the rare earth site. This agrees with
previous results on Ce$_3$Pt$_{23}$Si$_{11}$. For Pr- and
Nd$_3$Pt$_{23}$Si$_{11}$, the number of CEF parameters is too large to allow
for an unambiguous determination. This determination is possible for
Ce$_3$Pt$_{23}$Si$_{11}$, due to a reduced number of parameters and to the
availability of extensive experimental data. A specific procedure is developed
for this purpose that combines genetic algorithmics and optimization methods.
An unique set of CEF parameters is found for Ce$_3$Pt$_{23}$Si$_{11}$. It
reveals a strong anisotropy at the orthorhombic site, responsible for an easy
threefold magnetization axis in the cubic system. Using a microscopic,
mean-field, description, the magnetization processes in the paramagnetic and
ferromagnetic phases of Ce$_3$Pt$_{23}$Si$_{11}$ are well reproduced.
Ce$_3$Pt$_{23}$Si$_{11}$ is shown to realize a model for systems where
conflicting anisotropies are forced to cooperate.
