  A new general Lie-algebraic approach is proposed to solving evolution tasks
in some nonlinear problems of quantum physics with polynomially deformed Lie
algebras $su_{pd}(2)$ as their dynamic symmetry algebras.
  The method makes use of an expansion of the evolution operators by power
series in the $su_{pd}(2)$ shift operators and a (recursive) reduction of
finding coefficient functions to solving auxiliary exactly solvable $su(2)$
problems with quadratic Hamiltonians.
  PACS numbers: 03.70; 02.20; 42.50
