Shifted symplectic structures and Poisson vertex algebra

We identify Poisson vertex algebra (PVA) Hamiltonian structures on arc spaces with Maurer--Cartan data arising from degree-$1$ graded symplectic geometry. Let $X$ be a smooth scheme of finite type and set $Y=T^*[1]X$. We prove that a degree-$2$ Hamiltonian on $Y_\infty$ defines a sheaf of PVAs on $X_\infty$ precisely when its class in the Lie algebra of local functionals satisfies the classical master equation; conversely, every such PVA bracket determines a unique degree-$2$ local functional. We further show that classical $R$-matrices are exactly Maurer--Cartan deformations of the corresponding PVA Hamiltonian structure. As applications, we classify a class of global scalar PVA structures on $\mathbb P^1$ and apply the $R$-matrix formalism to finite and affine classical $\mathcal W$-algebras. For the subregular $\mathfrak{sl}_3$ example, we construct a compatible Poisson pencil and the centrally reduced RDW system, whose generic reduction is a genus-two hyperelliptic system admitting an explicit Mumford-type Lax representation. Its affine counterpart yields constant and first-order differential $R$-deformations, together with commuting Hamiltonian flows.