  In this paper we consider cases of existence of invariant measure, additional
first integrals, and Poisson structure in a problem of rigid body's rolling
without sliding on plane and sphere. The problem of rigid body's motion on
plane was studied by S.A. Chaplygin, P. Appel, D. Korteweg. They showed that
the equations of motion are reduced to a second-order linear differential
equation in the case when the surface of dynamically symmetric body is a
surface of revolution. These results were partially generalized by P. Woronetz,
who studied the motion of body of revolution and the motion of round disk with
sharp edge on the surface of sphere. In both cases the systems are Euler-Jacobi
integrable and have additional integrals and invariant measure. It turns out
that after some change of time defined by reducing multiplier, the reduced
system is a Hamiltonian system. Here we consider different cases when the
integrals and invariant measure can be presented as finite algebraic
expressions.
  We also consider the generalized problem of rolling of dynamically
nonsymmetric Chaplygin ball. The results of studies are presented as tables
that describe the hierarchy of existence of various tensor invariants:
invariant measure, integrals, and Poisson structure in the considered problems.
