  We give a characterization of linear canonoid transformations on symplectic
manifolds and we use it to generate biHamiltonian structures for some
mechanical systems. Utilizing this characterization we also study the behavior
of the harmonic oscillator under canonoid transformations. We present a
description of canonoid transformations due to E.T. Whittaker, and we show that
it leads, in a natural way, to the modern, coordinate-independent definition of
canonoid transformations. We also generalize canonoid transformations to
Poisson manifolds by introducing Poissonoid transformations. We give examples
of such transformations for Euler's equations of the rigid body (on
$\mathcal{so}^\ast (3) $ and $ so^\ast (4)$) and for an integrable case of
Kirchhoff's equations for the motion of a rigid body immersed in an ideal
fluid. We study the relationship between biHamiltonian structures and
Poissonoid transformations for these examples. We analyze the link between
Poissonoid transformations, constants of motion, and symmetries.
