Backlund transformations for two new integrable partial differential equations

We give two new completely integrable sixth-order partial differential equations, together with their Lax pairs and Darboux transformations. From these last we are able to derive the corresponding Backlund transformations. A simple reduction of these partial differential equations yields an integrable fourth-order ordinary differential equation for which a linear problem was previously unknown. We give also a linear problem for this ordinary differential equation.