  A class of scattered linearized polynomials covering infinitely many field
extensions is exhibited. More precisely, the $q$-polynomial over $\mathbb
F_{q^6}$, $q \equiv 1\pmod 4$ described in arXiv:1906.05611, arXiv:1910.02278
is generalized for any even $n\ge6$ to an $\mathbb F_q$-linear automorphism
$\psi(x)$ of $\mathbb F_q^n$ of order $n$. Such $\psi(x)$ and some functional
powers of it are proved to be scattered. In particular this provides new
maximum scattered linear sets of the projective line $\mathrm{PG}(1,q^n)$ for
$n=8,10$. The polynomials described in this paper lead to a new infinite family
of MRD-codes in $\mathbb F_q^{n\times n}$ with minimum distance $n-1$ for any
odd $q$ if $n\equiv0\pmod4$ and any $q\equiv1\pmod4$ if $n\equiv2\pmod4$.
