Population III Host Candidates at $z\sim2$: Strong He II $\lambda1640$ and Absent UV Metal Lines in HETDEX Ly$α$ Emitters
Population III (Pop III) stars are expected to produce extremely hard ionizing spectra, yet direct evidence for their signature--strong narrow He II $\lambda1640$ emission with weak or absent metal lines--remains elusive. Although Pop III formation peaks at $z\gtrsim10$, models predict that nearly pristine gas pockets can survive to later times, making intermediate-redshift searches a probe of metal mixing across cosmic time. We search 109,545 high-confidence Ly$α$-emitting galaxies at $1.9<z<2.3$ in the Hobby--Eberly Telescope Dark Energy Experiment's database for systems with strong He II $\lambda1640$ and no detected UV metal lines, recovering eight candidates and a first-order, tentative comoving number density of $\sim30$ Gpc$^{-3}$. The eight-object stack yields a rest-frame He II equivalent width of $28.7\pm6.7$ Angstrom, He II/Ly$α=0.354\pm0.094$, and a resolution-corrected He II FWHM of $449\pm105$ km s$^{-1}$. We find no significant N V, C IV, or O III] emission; using the He II width to set the scale, we derive $3σ$ upper limits of N V/He II $<0.167$, C IV/He II $<0.128$, and O III]/He II $<0.140$. This combination of strong He II, elevated He II/Ly$α$, moderate line width, and weak metal lines is difficult to reproduce with metal-enriched stellar populations, AGN narrow-line regions, shocks, or classical Wolf--Rayet features. We interpret the sample as candidates for very metal-poor or Pop III-like ionizing sources at $z\sim2$. Definitive confirmation requires deeper rest-UV spectroscopy to measure He II and metal-line limits in individual objects, followed by rest-optical spectroscopy of H$β$, [O III] $\lambda4959,\lambda5007$, [O II] $\lambda3727$, and H$α$ to constrain gas-phase metallicity, ionization conditions, and AGN activity.