  The maximum scattered linear sets in $PG(1,q^n)$ have been completely
classified for $n \le 4$ by Csajb\'ok-Zanella and Lavrauw-Van de Voorde. Here a
wide class of linear sets in $PG(1,q^5)$ is studied which depends on two
parameters. Conditions for the existence, in this class, of possible new
maximum scattered linear sets in $PG(1,q^5)$ are exhibited.
