  We obtain iso-spectral Euler-Bernoulli beams by using factorization and Lie
symmetry techniques. The canonical Euler-Bernoulli beam operator is factorized
as the product of a second-order linear differential operator and its adjoint.
The factors are then reversed to obtain iso-spectral beams. The factorization
is possible provided the coefficients of the factors satisfy a system of
non-linear ordinary differential equations . The uncoupling of this system
yields a single non-linear third-order ordinary differential equation. This
ordinary differential equation, refer to as the {\it principal equation}, is
analyzed and solved using Lie group methods. We show that the principal
equation may admit a one-dimensional or three-dimensional symmetry Lie algebra.
When the principal system admits a unique symmetry, the best we can do is to
depress its order by one. We obtain a one-parameter family of solutions in this
case. The maximally symmetric case is shown to be isomorphic to a Chazy
equation which is solved in closed form to derive the general solution of the
principal equation.
