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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 polynomialequations and inequalities expressing rank and positive semi-definitenessconditions of the $P$-matrix, a real symmetric matrix determined by theintegrity basis. The choice of the basic set of $G$-invariant homogeneouspolynomials forming an integrity basis is not unique, so it is not unique themathematical description of the orbit space too. If $G$ is an irreduciblefinite reflection group, Saito et al. in 1980 characterized some special basicsets of $G$-invariant homogeneous polynomials that they called {\em flat}. Theyalso found explicitly the flat basic sets of invariant homogeneous polynomialsof all the irreducible finite reflection groups except of the two largestgroups $E_7$ and $E_8$. In this paper the flat basic sets of invarianthomogeneous polynomials of $E_7$ and $E_8$ and the corresponding $P$-matricesare determined explicitly. Using the results here reported one is able todetermine 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 andinequalities defining the orbit spaces of $E_7$ and $E_8$ relatively to a flatbasis or to any other integrity basis. The results here obtained may beemployed concretely to study analytically the symmetry breaking in all theorieswhere 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.

Flat Bases of Invariant Polynomials and P-matrices of E7 and E8