  Symmetries and solutions of the Painleve IV equation are presented in an
alternative framework which provides the bridge between the Hamiltonian
formalism and the symmetric Painleve IV equation. This approach originates from
a method developed in the setting of pseudo-differential Lax formalism
describing AKNS hierarchy with the Darboux-Backlund and Miura transformations.
  In the Hamiltonian formalism the Darboux-Backlund transformations are
introduced as maps between solutions of the Hamilton equations corresponding to
two allowed values of Hamiltonian's discrete parameter. The action of the
generators of the extended affine Weyl group of the $A_2$ root system is
realized in terms of three "square-roots" of such Darboux-Backlund
transformations defined on a multiplet of solutions of the Hamilton equations.
