more general definition, see also [@mark]). For $0 < k < n$ it is a partial linear space. For $k=1$ and $k=n-1$ it is a projective space. So we assume that $$1< k < n-1.$$

It is known that there are two classes of maximal strong subspaces in $\M$: *stars* of the form $$\starof(H) = [H)_k = \{U\in\Sub_k(V)\colon H\subset U\},$$ where $H\in\Sub_{k-1}(V)$, and *tops* of the form $$\topof(B) = (B]_k = \{U\in\Sub_k(V)\colon U\subset B\},$$ where $B\in\Sub_{k+1}(V)$. Although non-maximal stars $[H, Y]_k$ and non-maximal tops $[Z, B]_k$, for some $Y, Z\in\Sub(V)$, make sense but in this paper when we say ‘a star’ or ‘a top’ we mean a maximal strong subspace. It is trivial that every line, a $k$-pencil $p=\penc(H, B)$, of $\M$ can be uniquely extended to the star $\starof(p):=\starof(H)$ and to the top $\topof(p):=\topof(B)$.

Numerous intrinsics properties of Grassmann spaces are very well known or can be easily verified (cf. [@mark]). Let us recall some of them and prove a couple of technical facts that will be needed later.

\[fact:lineinstartop\] Let $S$ be a star and $T$ be a top in $\M$. If $S\cap T\neq\emptyset$, then $S\cap T$ is a line and if additionally $U_1\in S\setminus T$, $U_2\in
  T\setminus 