The theory of $(k+1)$-Lagrangian spaces has been investigated in several papers as [1, 4]. In this paper an adapted frame in a $(k+1)$-Lagrangian space is chosen for an $f(2t+1,-1)$-structure, and matricies of the tensors $g$ and $f$ with respect to this adapted frame are obtained for $t=2^k$. Given is the necessary and sufficient condition for the $(k+1)$-Lagrangian space $E$ to admit a tensor field $f$ of type $(1,1)$ and rank $f=r=(k+1)\cdot n$, such that $f^{2\cdot2^k+1}-f=0$, $f^{2i+1}-f\neq0$ for $1\leq i<2^k$, $k\in N$.