  Let $M$ be an odd-dimensional Euclidean space endowed with a contact 1-form
$\alpha$. We investigate the space of symmetric contravariant tensor fields on
$M$ as a module over the Lie algebra of contact vector fields, i.e. over the
Lie subalgebra made up by those vector fields that preserve the contact
structure. If we consider symmetric tensor fields with coefficients in tensor
densities, the vertical cotangent lift of contact form $\alpha$ is a contact
invariant operator. We also extend the classical contact Hamiltonian to the
space of symmetric density valued tensor fields. This generalized Hamiltonian
operator on the symbol space is invariant with respect to the action of the
projective contact algebra $sp(2n+2)$. The preceding invariant operators lead
to a decomposition of the symbol space (expect for some critical density
weights), which generalizes a splitting proposed by V. Ovsienko.
