Algebraic solutions of the Hirota bilinear form for the Korteweg-de Vries and Boussinesq equations

We reduce the order of the fourth-order Hirota-bilinear equations of the Korteweg-de Vries and Boussinesq equations by using symmetries and then apply singularity analysis to the resultant ordinary differential equations. We employ all the possible combinations of symmetries to reduce the order of the equations.