One can extend the Fukaya-Oh category considering Lagrangian submanifolds in $X^{\vee}$ which are not necessarily unramified coverings of $Y$. For example, one can try to add to $FO(X^{\vee})$ new objects which are local systems on Lagrangian tori which are fibers of the projection $p^{\vee}:X^{\vee}\to Y$. It seems that with these objects one can go much further than with transversal ones. For example, in the general case of torus fibrations with singular fibers, one can argue that for almost any $y\in Y$ there is no limiting holomorphic discs with the boundary in the torus $(p^{\vee})^{-1}(y)$ . The set of such points $y$ is the complement to a countable union $Z$ of hypersurfaces in $Y$ (this follows from the fact that $dim(Y^{sing})=dim(Y)-2$). Thus, we get a large collection of honest objects without the parasitic composition $m_{0}$. The total picture seems to be quite intricate, as examples show that the subset $Z$ is everywhere dense. Presumably, it is related with some mysterious non-abelian $1$-cocycle which we will discuss later in the remark in section 7.1.