  Let G be an exceptional simple algebraic group, and let T be a maximal torus
in G. In this paper, for every such G, we find all simple rational G-modules V
with the following property: for every vector v in V, the closure of its
T-orbit is a normal affine variety. For all G-modules without this property we
present a T-orbit with the non-normal closure. To solve this problem, we use a
combinatorial criterion of normality formulated in the terms of weights of a
simple G-module. This paper continues two papers of the second author, where
the same problem was solved for classical linear groups.
