SYMMETRIES AND SIMILARITY REDUCTIONS OF THE DYM EQUATION

The eigenfunctions of the recursion operator Phi with eigenvalue lambd a(i), and the inverse of the recursion operator Phi(i) = Phi - lambda( i) for the Dym equation (DE) are obtained by means of the pseudopotent ials or the isospectral functions of the isospectral problem. The symm etries and algebras of the DE are also given. Using some symmetry suba lgebra of the DE, thirteen types of the significant similarity reducti ons are obtained by virtue of the classical Lie approach. For six type s of reductions, the general solutions can be obtained by means of the Weierstrass elliptic function, Riemann's zeta function, Jacobi ellipt ic functions and the solutions of a Riccati equation implicitly. Three types of reductions can all be solved by means of the Painleve II equ ation but with different independent arguments. Some types of nontrave lling singular solitary waves exist for the Dym equation there are two types of nonsingular nontravelling solitary wave solutions for some s uitable potentials.