= Solution
The area form $\sigma$ on $S^2$ is closed. For any closed two-form $B$ on a base, the <twisted cotangent symplectic form> $\omega+\pi^*B$ is closed and nondegenerate: pairing a putative kernel vector first with vertical vectors kills its horizontal component, and then pairing with horizontal vectors kills its vertical component. Thus both $\omega_+$ and $\omega_-$ are symplectic.
The cotangent bundle deformation-retracts onto its zero section. Their cohomology classes satisfy
$$
[\omega_+]-[\omega_-]=2\pi^*[\sigma]\ne0\in H^2(T^*S^2;\mathbb R).
$$
They are therefore not <strong isotopy of symplectic forms>[strongly isotopic].
Let $f:S^2\to S^2$ be an orientation-reversing isometry. Then $f^*\sigma=-\sigma$, while its cotangent lift preserves the canonical form and satisfies $\pi\circ f_\#=f\circ\pi$. Consequently
$$
f_\#^*\omega_+=\omega+\pi^*f^*\sigma=\omega_-,
$$
so the two forms are <symplectomorphic>.
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