iso-spectral Euler-Bernoulli beams \`a la Sophus Lie

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.