  An exact Calabi-Yau structure, originally introduced by Keller, is a special
kind of smooth Calabi-Yau structure in the sense of Kontsevich-Vlassopoulos.
For a Weinstein manifold $M$, the existence of an exact Calabi-Yau structure on
the wrapped Fukaya category $\mathcal{W}(M)$ imposes strong restrictions on its
symplectic topology. Under the cyclic open-closed map constructed by Ganatra,
an exact Calabi-Yau structure on $\mathcal{W}(M)$ induces a class $\tilde{b}$
in the degree one equivariant symplectic cohomology $\mathit{SH}_{S^1}^1(M)$.
Any Weinstein manifold admitting a quasi-dilation in the sense of
Seidel-Solomon has an exact Calabi-Yau structure on $\mathcal{W}(M)$. We prove
that there are many Weinstein manifolds whose wrapped Fukaya categories are
exact Calabi-Yau despite the fact the fact there is no quasi-dilation in
$\mathit{SH}^1(M)$, a typical example is given by the affine hypersurface
$\{x^3+y^3+z^3+w^3=1\}\subset\mathbb{C}^4$. As an application, we prove the
homological essentiality of Lagrangian spheres in many odd-dimensional smooth
affine varieties with exact Calabi-Yau wrapped Fukaya categories.
