and only if there exists a diffeomorphism ${{\varphi}}$ of $M$, inducing the identity on homology, such that ${{\varphi}}^*{{\omega}}_1={{\omega}}_0$.*]{}

In particular, questions 2 and 3 have positive answers. The proofs rely on the following observations.

[**1.**]{} For every $c\in{{\mathcal C}}$ there exist precisely three classes $E^c_1,E^c_2,E^c_3\in{{\mathcal E}}$ such that ${\langle c, E^c_i\rangle} = 1$. The numbering can be chosen such that $$\label{eq:Ec}
E^c_1\cdot E^c_3 = E^c_2\cdot E^c_3=1,\qquad 
E^c_1\cdot E^c_2 = 0.$$ Each class $E^c_i$ is represented by an embedded sphere. For the standard first Chern class $c_{{\mathrm{std}}}={{\mathrm{PD}}}(3L-E_1-E_2)\in{{\mathcal C}}$ these can be chosen as the classes $E^{c_{{\mathrm{std}}}}_1=E_1$, $E^{c_{{\mathrm{std}}}}_2=E_2$, $E^{c_{{\mathrm{std}}}}_3=L-E_1-E_2$. For general elements $c\in{{\mathcal C}}$ this follows from the fact that, by the results of Wall [@WALL1; @WALL2; @WALL3; @WALL4], the diffeomorphism group acts transitively on ${{\mathcal C}}$. (See Li–Liu [@LL1; @LL2] for details.)

[**2.**]{} By Taubes–Seiberg–Witten theory [@TAUBES3; @TAUBES4; @TAUBES5] the homology classes $E^c_1,E^c_2,E^c_3$ are represented by symplectically embedded spheres for every symplectic form on $M$ with first Chern class $c$.

[**3.**]{} A cohomology class $a\in H^2(M;{{\mathbb R}})$ can be represented by a symplectic form ${{\omega}}$ with first Chern class $c_1({{\omega}})=c$ if and only if for every $E\in H_2(M;{{\mathbb Z}})$ $$\label{eq:a-symp}
E\cdot E=-1,\quad {\langle c, E\rangle}=1
\qquad\implies\qquad {\langle a, E\rangle}>0.$$ By 2. 