Let us suppose that our collection of lines satisfies the following assumptions:

Assumption A1

There is a smooth metric $g=g_{B}$ and a collection of balls $D(s,r_{s})$ with centers at $s\in B^{sing}$ such that each ball $D(s,r_{s})$ contains exactly two lines $l_{\pm}\in{\cal L}_{in}$ outcoming of $s$.

Assumption A2

There exists $\varepsilon>0$ such that for any $p=f_{l}(t)\in Y^{\prime}:=B\setminus\cup_{s\in B^{sing}}D(s,r_{s})$ the distance in $T_{p}Y$ between $0\in T_{p}Y$ and the boundary of $P_{l,t}$ is greater or equal to $\varepsilon$.

We are going to show that such a collection does exist in Section 11.

Assumptions A1 and A2 are very artificial, they do not hold in physical picture which is the main motivation for the construction. It is quite possible that they can be weakened or even omitted. The main purpose of introducing them here is the possibility to define the sheaf of analytic functions by simple gluing. In complex geometry it is similar to the gluing of closed Riemann surfaces with boundaries by the mean of real-analytic identifications of the boundaries. It is well-known that one can replace real-analytic maps by smooth ones (or even by quasi-symmetric continuous maps). Maybe the rest of this section is unnecessary, and unpleasant technical arguments in Section 11.5 can be avoided.