  We completely solve the equivalence problem for Euler-Bernoulli equation
using Lie symmetry analysis. We show that the quotient of the symmetry Lie
algebra of the Bernoulli equation by the infinite-dimensional Lie algebra
spanned by solution symmetries is a representation of one of the following Lie
algebras: $2A_1$, $A_1\oplus A_2$, $3A_1$, or $A_{3,3}\oplus A_1$. Each
quotient symmetry Lie algebra determines an equivalence class of
Euler-Bernoulli equations. Save for the generic case corresponding to arbitrary
lineal mass density and flexural rigidity, we characterize the elements of each
class by giving a determined set of differential equations satisfied by
physical parameters (lineal mass density and flexural rigidity). For each
class, we provide a simple representative and we explicitly construct
transformations that maps a class member to its representative. The maximally
symmetric class described by the four-dimensional quotient symmetry Lie algebra
$A_{3,3}\oplus A_1$ corresponds to Euler-Bernoulli equations homeomorphic to
the uniform one (constant lineal mass density and flexural rigidity) . We
rigorously derive some non-trivial and non-uniform Euler-Bernoulli equations
reducible to the uniform unit beam. Our models extend and emphasize the
symmetry flavor of Gottlieb's iso-spectral beams (Proceedings of the Royal
Society London A 413 (1987) 235-250)
