In this paper was chose an adapted frame for $f(2k+1,-1)$-structure and a matrix of tensors $g_{ij}$ and $f^j_i$ with respect to this adapted frame. Given is the necessary and sufficient condition for an $n$-dimensional manifold $M^n$ to admit a tensor field $f$ of the type $(1,1)$ and the rank $r$ such that $f^{2\cdot2^q+1}-f=0$, $f^{2i+1}-f\neq0$ for $1\leq i<2^q$, $q\in N$.