Here we offer a construction of the set of lines satisfying the above axioms.

Let us use the standard ${\bf R}^{2}$ as a model around each $b\in B^{sing}$ in order to fix a structure of smooth manifold on the whole surface $B$. Let $\widetilde{Y}$ denotes the covering of $Y$ such that the fiber over $y\in Y$ is $(T_{y}^{\ast}Y)^{{\bf Z}}\setminus\{0\}$.

Let us fix a generic smooth metric on $B$. By the pull-back it gives a metric on $\widetilde{Y}$. Notice that there is a canonical closed $1$-form $\beta$ on $\widetilde{Y}$ such that $\beta_{|(y,\mu)}=\mu$, where $y\in Y,\,\,\mu\in(T_{y}^{\ast}Y)^{{\bf Z}}$. Using the metric we obtain dual to $\beta$ gradient vector field $v$ on $\widetilde{Y}$.

For any $s\in B^{sing}$ and a choice of 1-form $\alpha(0)=\pm dy$ in local coordinates, we take the unique integral line of $v$ starting at $(s,\alpha(0))$. Set ${\cal L}_{in}$ will be the set of all lines obtained in this way. Each line $l\in{\cal L}_{in}$ carries a covariantly constant closed $1$-form $\alpha_{l}$. Using Axiom 2 as a definition, we obtain a canonical parametrization of each line by the time parameter $t$. Since the metric is generic, a line cannot return to a point in $B^{sing}$.

Then we proceed inductively. If two already constructed lines $l_{1},l_{2}\in{\cal L}$ meet at $x\in Y$ we produce a new integral line $l$ of $v$ with the direction covector satisfying the condition 3b) for any pair of coprime positive integers $n_{1},n_{2}$. In this way we construct a set of lines ${\cal L}$ satisfying all the axioms. The only non-trivial thing to check is that for each line values of the parameter $t$ are in one-to-one correspondence with the interval $(0,+\infty)$. In order to see this we observe that the length of each line is infinite. Indeed, an integral curve of $v$ cannot have a limiting point in $Y$ (since the flow generated by $v$ is smooth, and the lengths of tangent vectors are bounded from below because of the integrality of $1$-forms).

We conclude that there exists a set ${\cal L}$ of lines satisfying Axioms 1-7.