A note on the integrability of a remarkable static Euler-Bernoulli beam equation

It has recently been shown that the fourth-order static Euler-Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, in the maximal case has three symmetries. This corresponds to the negative fractional power law y (-5/3), and the equation has the nonsolvable algebra . We obtain new two- and three-parameter families of exact solutions when the equation has this symmetry algebra. This is studied via the symmetry classification of the three-parameter family of second-order ordinary differential equations that arises from the relationship among the Noether integrals. In addition, we present a complete symmetry classification of the second-order family of equations. Hence the admittance of remarkably allows for a three-parameter family of exact solutions for the static beam equation with load a fractional power law y (-5/3).