  A spin-U(1)-symmetry protected momentum-dependent integer-$Z$-valued
topological invariant is proposed to time-reversal-invariant (TRI)
superconductivity (SC) whose nonzero value will lead to exactly flat surface
band(s). The theory is applied to the weakly spin-orbit coupled quasi-1D
A$_2$Cr$_3$As$_3$ (A=Na, K, Rb, Cs) superconductors family with highest $T_c$
up to 8.6 K with $p_z$-wave pairing in the $S_z=0$ channel. It's found that up
to the leading atomic spin-orbit-coupling (SOC), the whole (001) surface
Brillouin zone is covered with exactly-flat surface bands, with some regime
hosting three flat bands and the remaining part hosting two. Such exactly-flat
surface bands will lead to very sharp zero-bias conductance peak in the
scanning tunneling microscopic spectrum. When a tiny subleading spin-flipping
SOC is considered, the surface bands will only be slightly split. The
application of this theory can be generalized to other unconventional
superconductors with weak SOC, particularly to those with mirror-reflection
symmetry.
