  Let $G$ be a compact group of linear transformations of an Euclidean space
$V$. The $G$-invariant $C^\infty$ functions can be expressed as $C^\infty$
functions of a finite basic set of $G$-invariant homogeneous polynomials,
called an integrity basis. The mathematical description of the orbit space
$V/G$ depends on the integrity basis too: it is realized through polynomial
equations and inequalities expressing rank and positive semi-definiteness
conditions of the $P$-matrix, a real symmetric matrix determined by the
integrity basis. The choice of the basic set of $G$-invariant homogeneous
polynomials forming an integrity basis is not unique, so it is not unique the
mathematical description of the orbit space too. If $G$ is an irreducible
finite reflection group, Saito et al. in 1980 characterized some special basic
sets of $G$-invariant homogeneous polynomials that they called {\em flat}. They
also found explicitly the flat basic sets of invariant homogeneous polynomials
of all the irreducible finite reflection groups except of the two largest
groups $E_7$ and $E_8$. In this paper the flat basic sets of invariant
homogeneous polynomials of $E_7$ and $E_8$ and the corresponding $P$-matrices
are determined explicitly. Using the results here reported one is able to
determine easily the $P$-matrices corresponding to any other integrity basis of
$E_7$ or $E_8$. From the $P$-matrices one may then write down the equations and
inequalities defining the orbit spaces of $E_7$ and $E_8$ relatively to a flat
basis or to any other integrity basis. The results here obtained may be
employed concretely to study analytically the symmetry breaking in all theories
where the symmetry group is one of the finite reflection groups $E_7$ and $E_8$
or one of the Lie groups $E_7$ and $E_8$ in their adjoint representations.
