  We study integrable systems on double Lie algebras in absence of Ad-invariant
bilinear form by passing to the semidirect product with the $\tau
$-representation. We show that in this stage a natural Ad-invariant bilinear
form does exist, allowing for a straightforward application of the AKS theory,
and giving rise to Manin triple structure, thus bringing the problem to the
realm of Lie bialgebras and Poisson-Lie groups.
