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Symmetric tensors on the intersection of two quadrics and Lagrangian fibration
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Let
$X$ be an 
$n$-dimensional (smooth) intersection of two quadrics, and let 
${T^{\rm{*}}}X$ be its cotangent bundle. We show that the algebra of symmetric tensors on 
$X$ is a polynomial algebra in 
$n$ variables. The corresponding map 
${\rm{\Phi }}:{T^{\rm{*}}}X \to {\mathbb{C}^n}$ is a Lagrangian fibration, which admits an explicit geometric description; its general fiber is a Zariski open subset of an abelian variety, which is a quotient of a hyperelliptic Jacobian by a 
$2$-torsion subgroup. In dimension 
$3$, 
${\rm{\Phi }}$ is the Hitchin fibration of the moduli space of rank 
$2$ bundles with fixed determinant on a curve of genus 
$2$.
