In the present paper results about the "Generalized Polynomial Korteweg-de Vries equation" (GPKdV) are obtained, extending the ones by Sachs [SIAM J. Math. Anal. 14, 674 (1983)] for the Korteweg-de Vries (KdV) equation. Namely, the evolution of the so-called "prolonged squared" eigenfunctions of the associated spectral problem according to the linearized GPKdV is proven, the Lax pairs associated with the "prolonged" eigenfunctions as well as "prolonged squared" eigenfunctions are derived, and on the basis of some expansion formulas the Cauchy problem for the linearized GPKdV with a decreasing at infinity initial condition is solved.