A Lagrangian neighborhood germ is not uniquely determined

Already on $T^{\ast }\mathbb{R}={\mathbb{R}}^2$ with coordinates $(q,p)$, the map $F(q,p)=(q+p,p)$ is a nonidentity diffeomorphism germ along the zero section. It fixes every $(q,0)$ and satisfies $F^{\ast }(dq\wedge dp)=d(q+p)\wedge dp=dq\wedge dp$.

For this cotangent model, the identity map is one symplectomorphism germ fixing the zero section, and the shear $F$ from step 1.1 is another. They are distinct and have the same restriction to every point of that section. Hence these data do not uniquely determine a germ. The explicit pair requires no existence theorem or choice principle and makes no separate claim that a natural distinguished choice is impossible.