The reduced self-dual Yang–Mills equation, binary and infinitesimal Darboux transformations

The self-dual Yang–Mills equation in the (2+1)-dimensional space–time is considered. Binary Darboux transformation of a new kind is applied to obtain the infinite hierarchy of solutions expressed through ones of corresponding Lax pairs on an initial solution of the equation studied. Sufficient conditions are given for the solutions built to take values in the class of unimodular positive-definite matrices. A new infinitesimal Darboux transformation is introduced and a particular solution of the linearization of the self-dual Yang–Mills equation is given. Two hierarchies of infinitesimal symmetries depending on a solution of the nonlinear equation are extracted.