  Motivated by the Beauville decomposition of an abelian scheme and the
"Perverse = Chern" phenomenon for a compactified Jacobian fibration, we study
in this paper splittings of the perverse filtration for compactified Jacobian
fibrations.
  On the one hand, we prove for the Beauville-Mukai system associated with an
irreducible curve class on a K3 surface the existence of a Fourier-stable
multiplicative splitting of the perverse filtration, which extends the
Beauville decomposition for the nonsingular fibers. Our approach is to
construct a Lefschetz decomposition associated with a Fourier-conjugate
$\mathfrak{sl}_2$-triple, which relies heavily on recent work concerning the
interaction between derived equivalences and LLV algebras for hyper-K\"ahler
varieties. Motivic lifting and connections to the Beauville-Voisin conjectures
are also discussed.
  On the other hand, we construct for any $g\geq 2$ a compactified Jacobian
fibration of genus g curves such that each curve is integral with at worst
simple nodes and the (multiplicative) perverse filtration does not admit a
multiplicative splitting. This shows that in general an extension of the
Beauville decomposition cannot exist for compactified Jacobian fibrations even
when the simplest singular point appears.
