  In this paper, we study the implications of the Dark Large Mixing Angle
(DLMA) solutions of $\theta_{12}$ in the context of the IceCube data. We study
the consequences in the measurement of the neutrino oscillation parameters
namely $\theta_{23}$ and $\delta_{\rm CP}$ in light of both Large Mixing Angle
(LMA) and DLMA solutions of $\theta_{12}$. We find that it will be impossible
for IceCube to determine the $\delta_{\rm CP}$ and the true nature of
$\theta_{12}$ i.e., LMA or DLMA at the same time. This is because of the
existence of an intrinsic degeneracy at the Hamiltonian level between these
parameters. Apart from that, we also identify a new degeneracy between
$\theta_{23}$ and two solutions of $\theta_{12}$ for a fixed value of
$\delta_{\rm CP}$. We perform a chi-square fit using three different
astrophysical sources, i.e., $\mu$ source, $\pi$ source, and $n$ source to find
that both $\mu$ source and $\pi$ source are allowed within $1 \sigma$ whereas
the $n$ source is excluded at $2 \sigma$. It is difficult to make any
conclusion regarding the measurement of $\theta_{23}$, $\delta_{\rm CP}$ for
$\mu$ source. However, The $\pi$ ($n$) source prefers higher (lower) octant of
$\theta_{23}$ for both LMA and DLMA solution of $\theta_{12}$. The best-fit
value of $\delta_{\rm CP}$ is around $180^\circ$ ($0^\circ/360^\circ$) for LMA
(DLMA) solution of $\theta_{12}$ whereas for DLMA (LMA) solution of
$\theta_{12}$, the best-fit value is around $0^\circ/360^\circ$ ($180^\circ$)
for $\pi$ ($n$) source. If we assume the current best-fit values of
$\theta_{23}$ and $\delta_{\rm CP}$ to be true, then the $\mu$ and $\pi$ source
prefer the LMA solution of $\theta_{12}$ whereas the $n$ source prefers the
DLMA solution of $\theta_{12}$.
