Let A and B be two points of $$\mathop {\mathrm{PG}}(d,q^n)$$ PG(d,qn) and let $$\Phi $$ Φ be a collineation between the stars of lines with vertices A and B, that does not map the line AB into itself. In this paper we prove that if $$d=2$$ d=2 or $$d\ge 3$$ d≥3 and the lines $$\Phi ^{-1}(AB), AB, \Phi (AB) $$ Φ-1(AB),AB,Φ(AB) are not in a common plane, then the set $$\mathcal{C}$$ C of points of intersection of corresponding lines under $$\Phi $$ Φ is the union of $$q-1$$ q-1 scattered $${\mathbb {F}}_{q}$$ Fq -linear sets of rank n together with $$\{A,B\}$$ {A,B} . As an application we will construct, starting from the set $$\mathcal{C}$$ C , infinite families of non-linear $$(d+1, n, q;d-1)$$ (d+1,n,q;d-1) -MRD codes, $$d\le n-1$$ d≤n-1 , generalizing those recently constructed in Cossidente et al. (Des Codes Cryptogr 79:597–609, 2016) and Durante and Siciliano (Electron J Comb, 2017).