In a QM introductory book , I have read that the wave packet is not a solution of the Schrodinger equation, is this true in some context or is it just an mistake of the author?
It depends on the potential. If there is a potential, then the wave packet is not a solution. If there is no potential, then the Hamiltonian describes a free field, in which case a wave packet is a solution, and the Schrodinger equation describes how the packet moves and spreads out through time. The easiest way to see this is to break the wave packet apart into different energy eigenstates (this is just a Fourier expansion), which are trivial solutions of the SE.
Why? The Schrödinger equation is linear also in the presence of a potential, so you can still create a wave packet solution by superposing an infinity of solutions (all solutions if the system with potential of course).