  The local stellar halo of the Milky Way contains the debris from several past
accretion events. Here we study in detail the structure and properties of
nearby debris associated with the Helmi streams, originally identified as an
overdensity in integrals of motion space. We use 6D phase-space information
from Gaia EDR3 combined with spectroscopic surveys, and we analyse the orbits
and frequencies of the stars in the streams using various Galactic potentials.
We also explore how the Helmi streams constrain the flattening q, of the
Galactic dark matter halo. We find that the streams are split into
substructures in integrals of motion space, most notably into two clumps in
angular momentum space. The clumps have consistent metallicity distributions
and stellar populations, supporting a common progeny. In all the realistic
Galactic potentials explored, the Helmi streams stars depict a diffuse
distribution close to Omega_z /Omega_R ~0.7. At the same time, the reason for
the substructure in angular momentum space appears to be a Omega_z : Omega_phi
resonance close to the 1:1. This resonance is exactly the 1:1 in the case that
the (density) flattening of the dark halo is q = 1.2. For this halo shape the
substructure in angular momenta is also long lasting. Our findings suggest that
the structure of the Galactic potential leaves a clear imprint on the
properties of phase-mixed debris streams.
