Series Solution of the 1 + 2 Continuous Toda Chain

The 1+2 dimensional continuous Toda chain presents a formidable challenge to the construction of solutions. Two variable reductions of the equation are known, but up till now nothing more is known. In this paper a way to solve the equation is presented, and some solutions are explicitly constructed. The method depends upon connecting series solutions of the symmetry equation of the Toda chain with the Toda solutions, as is guaranteed by general theory. Such solutions of the symmetry equation are obtainable a series method. An explicit solution is constructed by this method, and a general procedure is given to realize further solutions, which, however are given only in implicit form.