Top panels: fiducial orbit for HE 0144−4657 (blue circle) in Cartesian Galactocentric coordinates within the P. J. McMillan (2017) model potential. Top left: (X, Z); Milky Way (MW) edge-on view. Top right: (X, Y); face-on. Blue arrows exhibit the present-day velocity vector of HE 0144−4657. Orange and green lines are forward and backward trajectories, respectively. Thicker lines show 200 Myr integration, while thinner transparent lines represent 2 Gyr total. Black dashed lines are the orbit of the Sun with the same assumptions (Section 3.3), and the black symbol marks its current Galactic location. The gray regions illustrate the MW’s thin disk with 18 kpc radius and 1 kpc maximum height. The thicker portion in the top-left panel is the thick disk (14 kpc radius and 3 kpc height). Bottom panels: dynamical parameter spaces. Bottom left: (L_z, E). Bottom center: projected action space (or action “diamond”). The horizontal axis displays L_z/J_total ∈ [ − 1, + 1], where J_total = J_R + ∣L_z∣ + J_z (see text). Vertical axis shows (J_z − J_R)/J_total ∈ [ − 1, + 1]. Bottom right: (L_z, L_⊥), where ﹩{L}_{\perp }=\sqrt{{L}_{x}^{2}+{L}_{y}^{2}}﹩. The white dashed rectangles delineate the Helmi Stream kinematic locus. In all bottom panels, green, yellow, and pink dots are Helmi Stream members from the SEGUE survey (G. C. Myeong et al. 2018), a very metal-poor sample ([Fe/H]≤ − 1.8) including HES data (G. Limberg et al. 2021b), and Gaia DR3 Radial Velocity Spectrometer instrument (E. Dodd et al. 2023). We also highlight the dynamical volume occupied by the most relevant halo substructures, namely Gaia-Sausage/Enceladus (“GSE”; V. Belokurov et al. 2018; A. Helmi et al. 2018) and Sagittarius stream (“Sgr”; R. A. Ibata et al. 1994; S. R. Majewski et al. 2003). The colored background 2D histograms consist of the SEGUE/StarHorse catalog (G. Limberg et al. 2023; A. B. A. Queiroz et al. 2023).