Effect of origin poles on stability?
A system with simple distinct poles on the imaginary axis (and note that the origin is on the imaginary axis) and no poles in the right half-plane is called marginally stable. If you have poles with multiplicity greater than $1$ on the imaginary axis, or if there are poles in the right half-plane, then the system is unstable.

For discrete-time systems, the same is true if you replace "imaginary axis" by "unit circle".

Transients in marginally stable systems do not decay, but neither do they grow without bounds. In practice, we usually want to avoid marginally stable systems. An ideal integrator is an example of a marginally stable system.