  A systematic analysis of a continuous version of a binomial lattice,
containing a real parameter $\gamma$ and covering the Toda field equation as
$\gamma\to\infty$, is carried out in the framework of group theory. The
symmetry algebra of the equation is derived. Reductions by one-dimensional and
two-dimensional subalgebras of the symmetry algebra and their corresponding
subgroups, yield notable field equations in lower dimensions whose solutions
allow to find exact solutions to the original equation. Some reduced equations
turn out to be related to potentials of physical interest, such as the
Fermi-Pasta-Ulam and the Killingbeck potentials, and others. An instanton-like
approximate solution is also obtained which reproduces the Eguchi-Hanson
instanton configuration for $\gamma\to\infty$. Furthermore, the equation under
consideration is extended to $(n+1)$--dimensions. A spherically symmetric form
of this equation, studied by means of the symmetry approach, provides
conformally invariant classes of field equations comprising remarkable special
cases. One of these $(n=4)$ enables us to establish a connection with the
Euclidean Yang-Mills equations, another appears in the context of Differential
Geometry in relation to the socalled Yamabe problem. All the properties of the
reduced equations are shared by the spherically symmetric generalized field
equation.
