Parametric (soliton) solutions to integrable systems as a subclass of the solutions depending on a set of arbitrary functions

In this paper, the authors establish a new invariance property of integrable dynamical systems which allows one to construct a hierarchy of the solutions depending on a set of arbitrary functions. A concrete choice of these functions leads to the known soliton: solutions to the corresponding nonlinear PDE's. Starting Points: all the integrable systems under consideration admit the transformation is: θ implies bar θ ≡ sθ = F (θ, θ(1)i...θNt), sN ≠ 1. Here θ and bar θ are unknown functions (variables) satisfying the corresponding PDE, θvi ≡ ∂vθ/∂xvi. The integer N depends on a concrete form of the equation. There is a solution θ0 of the nonlinear system in equation which depends on a set of arbitrary functions of the argument. From the group point of view this circumstance is related with the fact that Lax equations possess an exact solution not only for diagonal case but also for solvable (diagonal + upper (lower) triangler matrices)