  In this letter, we present developments of the Hamiltonian approach to
problems of the probability distribution for a passive scalar in isotropic
turbulence, and also considers specific applications of the modified
Prelle-Singer procedure to turbulence models. The following key questions are
discussed and solved: what is the general dynamical structure of the resulting
scale equation permitted by passive scalar turbulence models? What are the
general requirements of the relations between canonical variables and the
canonical variabes representation for turbulence by using canonical variables?
It is shown that the existence of the Haniltonian representation in turbulence
is a privilege of only turbulence systems for which the variational principle
of least action is impossible The master equation of the probability
distribution of a passive scalar in isotropic turbulence can also be deduced
explicitly.
