Generically rational polynomials of quasi-simple type

A polynomial f∈k[x,y] is generically rational if the curve f=α is rational for almost all α∈k. It is of quasi-simple type if the generic member of the pencil Λ(f) generated by f has all rational points at infinity except for one non-rational point of degree a. We classify all generically rational polynomials of quasi-simple type with a=2.