  We study completely integrable systems attached to Takiff algebras
$\mathfrak{g}_N$, extending open Toda systems of split simple Lie algebras
$\mathfrak{g}$. With respect to Darboux coordinates on coadjoint orbits
$\mathcal{O}$, the potentials of the hamiltonians are products of polynomial
and exponential functions. General solutions for equations of motion for
$\mathfrak{g}_N$ are obtained using differential operators called jet
transformations. These results are applied to a $3$-body problem based on
$\mathfrak{sl}(2)$, and to an extension of soliton solutions for $A_\infty$ to
associated Takiff algebras. The new classical integrable systems are then
lifted to families of commuting operators in an enveloping algebra, solving a
Vinberg problem and quantizing the Poisson algebra of functions on
$\mathcal{O}$.
