Continuous approximation of binomial lattices

A systematic analysis of a continuous version of a binomial lattice, contai ning a real parameter gamma and covering the Toda field equation as gamma - -> infinity, is carried out in the framework of group theory. The symmetry algebra of the equation is derived. Reductions by one-dimensional and two-d imensional subalgebras of the symmetry algebra and their corresponding subg roups, yield notable field equations in lower dimensions whose solutions al low us to find exact solutions to the original equation. Some reduced equat ions turn out to be related to potentials of physical interest, such as the Fermi-Pasta-Ulam and the Killingbeck potentials, and others. An instantonl ike approximate solution is also obtained which reproduces the Eguchi-Hanso n instanton configuration for gamma --> infinity. Furthermore, the equation under consideration is extended to n + 1 dimensions. A spherically symmetr ic form of this equation, studied by means of the symmetry approach, provid es conformally invariant classes of field equations comprising remarkable s pecial cases. One of these (n = 4) enables us to establish a connection wit h the Euclidean Yang-Mills equations, another appears in the context of Dif ferential Geometry in relation to the so-called Yamabe problem. All the pro perties of the reduced equations are shared by the spherically symmetric ge neralized field equation.