Given a symplectic 6-manifold $(M,\omega)$ and an embedded symplectic 2-sphere $C\subset M$ whose normal bundle has the first Chern class -2. How to find on $M$ another closed 2-form $\eta$ which only degenerate along the 2-sphere $C$?
Here, the restriction on a neighborhood of $C$ is isomorphic to the pull-back the canonical symplectic form on $\mathbb{C}^{4}$ by the following map:$$L\oplus L\to \{w^{2}+x^{2}+y^{2}+z^{2}=0\}\subset\mathbb{C}^{4}$$ $$w=a+b',x=i(a-b'),y=a'-b,z=i(a'+b)$$
where $L$ is the complex line bundle over $\mathbb{C}P^{1}$ with the first Chern class -1, $a,b$ are the linear functionals on the first $L$, $a',b'$ are the linear functionals on the second $L$, and they satisfy $ab'-a'b=0$(see C. H. Clemens' paper "Double Solids"(Adv. in Math. 47(1983), 107-230))
In I. Smith, R.P. Thomas, S.-T. Yau's paper"Symplectic Conifold Transitions" (J. Differential Geometry 62(2002), 209-242) , it says that "finding families of symplectic forms which degenerate only along such spheres, yielding conifolds, is more subtle". According to their paper, finding such a 2-form $\eta$ is a necessary and sufficient condition which can make the corresponding symplectic surgery (maybe called "inverse symplectic conifold transition") realizable. I wondered if there is any other paper discussion about this topic until now.