  To some braiding R of Hecke type (a Hecke symmetry) we put into
correspondence an associative algebra called the modified Reflection Equation
Algebra (mREA). We construct a series of matrices L_(m), m=1,2,... with entries
belonging to mREA such that each of them satisfies a version of the
Cayley-Hamilton identity with central coefficients.
  We also consider some quotients of the mREA which are called the
non-commutative orbits. For each of these orbits we construct a large family of
projective modules. In this family we introduce an algebraic structure which is
close to that of $K^0(\Fl(\C^n))$. The algebraic structure respects an
equivalence relation motivated by a "quantum" trace compatible with the initial
Hecke symmetry R. For a subclass of non-commutative orbits we compute the
spectrum of central elements of the mREA Tr_R(L_(m)^k), k\in {\Bbb N}.
