  We study a number of supersymmetric solutions in the form of $Mkw_3\times
S^3$- and $AdS_3\times S^3$-sliced domain walls in the maximal gauged
supergravity in seven dimensions. These solutions require non-vanishing
three-form fluxes to support the $AdS_3$ and $S^3$ subspaces. We consider
solutions with $SO(4)$, $SO(3)$, $SO(2)\times SO(2)$ and $SO(2)$ symmetries in
$CSO(p,q,5-p-q)$, $CSO(p,q,4-p-q)$ and $SO(2,1)\ltimes \mathbf{R}^4$ gauge
groups. All of these solutions can be analytically obtained. For $SO(5)$ and
$CSO(4,0,1)$ gauge groups, the complete truncation ansatze in terms of
eleven-dimensional supergravity on $S^4$ and type IIA theory on $S^3$ are
known. We give the full uplifted solutions to eleven and ten dimensions in this
case. The solutions with an $AdS_3\times S^3$ slice are interpreted as
two-dimensional surface defects in six-dimensional $N=(2,0)$ superconformal
field theory in the case of $SO(5)$ gauge group or $N=(2,0)$ nonconformal field
theories for other gauge groups. For $SO(4)$ symmetric solutions, it is
possible to find solutions with both the three-form fluxes and $SO(3)$ gauge
fields turned on. However, in this case, the solutions can be found only
numerically. For $SO(3)$ symmetric solutions, the three-form fluxes and $SO(3)$
gauge fields cannot be non-vanishing simultaneously.
