Let $Y\subset X$ be a Lagrangian submanifold in a holomorphic symplectic manifold $X$. We know that there exists a local moduli space $M$, which parametrizes lagrangian submanifolds in $X$(there are no obstructions), and such that there is a natural isomorphism $T_{[Y]} \cong H^0(Y,N_{Y/X}) $. Let us fix a section $s\in H^0(Y,N_{Y/X})$. Suppose that $s$ is non-vanishing. Is it true that for a sufficiently small disk $U\subset \mathbb{C}$ there exists a holomorphic map $\gamma: U\to M$, such that $\gamma(0)=[Y]$ and $\frac{d\gamma}{dz}|_{z=0}=s$ and such that submanifolds $Y_{\gamma(t)}$ don't intersect each other? Here by $Y_{\gamma(t)}$ we denote the Lagragian submanifold in $X$ which corresponds to a point $\gamma(t)$ in the moduli space $M$ . Or more generally: if $s$ has zeros, is it true that $Y_{\gamma(t)}$ intersect each other only by zeros of $s$?

Let $Y\subset X$ be a Lagrangian submanifold in a holomorphic symplectic manifold $X$. We know that there exists a local moduli space $M$, which parametrizes lagrangian submanifolds in $X$(there are no obstructions), and such that there is a natural isomorphism $T_{[Y]} \cong H^0(Y,N_{Y/X}) $. Let us fix a section $s\in H^0(Y,N_{Y/X})$. Suppose that $s$ is nowhere vanishing. Is it true that for a sufficiently small disk $U\subset \mathbb{C}$ there exists a holomorphic map $\gamma: U\to M$, such that $\gamma(0)=[Y]$ and $\frac{d\gamma}{dz}|_{z=0}=s$ and such that submanifolds $Y_{\gamma(t)}$ don't intersect each other? Here by $Y_{\gamma(t)}$ we denote the Lagragian submanifold in $X$ which corresponds to a point $\gamma(t)$ in the moduli space $M$ . Or more generally: if $s$ has zeros, is it true that $Y_{\gamma(t)}$ intersect each other only by zeros of $s$?