  In this paper we show that the quantum theory of chaos, based on the
statistical theory of energy spectra, presents inconsistencies difficult to
overcome. In classical mechanics a system described by an hamiltonian $H = H_1
+ H_2$ (decomposable) cannot be ergodic, because there are always two dependent
integrals of motion besides the constant of energy. In quantum mechanics we
prove the existence of decomposable systems \linebreak $H^q = H^q_1 + H^q_2$
whose spacing distribution agrees with the Wigner law and we show that in
general the spacing distribution of $H^q$ is not the Poisson law, even if it
has often the same qualitative behaviour. We have found that the spacings of
$H^q$ are among the solutions of a well defined class of homogeneous linear
systems. We have obtained an explicit formula for the bases of the kernels of
these systems, and a chain of inequalities which the coefficients of a generic
linear combination of the basis vectors must satisfy so that the elements of a
particular solution will be all positive, i.e. can be considered a set of
spacings.
