  It is currently a big challenge to accurately determine the symmetry energy
$E_{\text{sym}}(\rho)$ and the pure neutron matter equation of state
$E_{\text{PNM}}(\rho)$, even their values around saturation density $\rho_0 $.
We find that the electric dipole polarizability $\alpha _ {\text{D}}$ in
$^{208}$Pb can be determined uniquely by the magnitude of the
$E_{\text{sym}}(\rho)$ or almost equivalently the $E_{\text{PNM}}(\rho)$ at
subsaturation densities around $\rho_0/3 $, shedding a light upon the genuine
correlation between the $\alpha _ {\text{D}}$ and the $E_{\text{sym}}(\rho)$.
By analyzing the experimental data of the $\alpha _ {\text{D}}$ in $^{208}$Pb
from RCNP using a number of non-relativistic and relativistic mean-field
models, we obtain very stringent constraints on $E_{\text{sym}}(\rho)$ and
$E_{\text{PNM}}(\rho)$ around $\rho_0/3 $. The obtained constraints are found
to be in good agreement with the results extracted in other analyses. In
particular, our results provide for the first time the experimental constraints
on $E_{\text{PNM}}(\rho)$ around $\rho_0/3 $, which are in harmony with the
recent determination of $E_{\text{PNM}}(\rho)$ from microscopic theoretical
studies and potentially useful in constraining the largely uncertain
many-nucleon interactions in microscopic calculations of neutron matter.
