  Low-temperature (down to $\sim$ 50 mK) thermal conductivity measurements with
the heat flow direction along the inter-plane tetragonal $c$-axis, $\kappa_c$,
were used to study the superconducting state of heavy fermion CeIrIn$_5$.
Measurements were performed in the magnetic fields both parallel to the heat
flow direction, $H\parallel c$, and transverse to it, $H \parallel a$.
Inter-plane heat conductivity in $H \parallel c$ configuration shows negligible
initial increase with magnetic field and a rapid rise on approaching $H_{c2}$
from below, similar to the expectations for the superconducting gap without
line nodes. This observation is in stark contrast to monotonic increase found
in the previous in-plane heat transport measurements. In the configuration with
the magnetic field breaking the tetragonal symmetry of the lattice, $H
\parallel a$, $\kappa_c$ reveals non-monotonic evolution with temperature and
magnetic field suggesting sub-phase boundary in the superconducting state. The
characteristic temperature $T_{kink} \sim$ 0.07~K of the sub-boundary is well
within the domain of bulk superconductivity $T_c \sim$ 0.4~K and $H_{c2}\sim$
1.0~T. These results are consistent with a superconducting gap with an
equatorial line node and polar point nodes, a gap symmetry of the D$_{4h}$
point group, for which magnetic field along the tetragonal plane breaks the
degeneracy of the multi-component order parameter and induces a phase
transition with nodal topology change.
