Actions of the n-strand braid groups on the free group of rank n which are similar to the Artin representation

We classify an action of the $n$-strand braid group on the free group of rank $n$ which is similar to the Artin representation in the sense that the $i$-th generator ${\sigma }_i$ of $B_n$ acts so that it fixes all free generators $x_j$ except $j=i,i+1$. We determine all such representations and discuss knot invariants coming from such representations.