  Strong high-ionization lines such as HeII of young galaxies are puzzling at
high and low redshift. Although recent studies suggest the existence of
non-thermal sources, whether their ionizing spectra can consistently explain
multiple major emission lines remains a question. Here we derive the general
shapes of the ionizing spectra for three local extremely metal-poor galaxies
(EMPGs) that show strong HeII$\lambda$4686. We parameterize the ionizing
spectra composed of a blackbody and power-law radiation mimicking various
stellar and non-thermal sources. We use photoionization models for nebulae, and
determine seven parameters of the ionizing spectra and nebulae by Markov Chain
Monte Carlo methods, carefully avoiding systematics of abundance ratios. We
obtain the general shapes of ionizing spectra explaining $\sim 10$ major
emission lines within observational errors with smooth connections from
observed X-ray and optical continua. We find that an ionizing spectrum of one
EMPG has a blackbody-dominated shape, while the others have convex downward
shapes at $>13.6$ eV, which indicate a diversity of the ionizing spectrum
shapes. We confirm that the convex downward shapes are fundamentally different
from ordinary stellar spectrum shapes, and that the spectrum shapes of these
galaxies are generally explained by the combination of the stellar and
ultra-luminous X-ray sources. Comparisons with stellar synthesis models suggest
that the diversity of the spectrum shapes arises from differences in the
stellar age. If galaxies at $z\gtrsim 6$ are similar to the EMPGs, high energy
($>54.4$ eV) photons of the non-stellar sources negligibly contribute to cosmic
reionization due to relatively weak radiation.
