  In this paper, subsystems with shadowing property for
$\mathbb{Z}^{k}$-actions are investigated. Let $\alpha$ be a continuous
$\mathbb{Z}^{k}$-action on a compact metric space $X$. We introduce the notions
of pseudo orbit and shadowing property for $\alpha$ along subsets, particularly
subspaces, of $\mathbb{R}^{k}$. Combining with another important property
"expansiveness" for subsystems of $\alpha$ which was introduced and
systematically investigated by Boyle and Lind, we show that if $\alpha$ has the
shadowing property and is expansive along a subspace $V$ of $\mathbb{R}^{k}$,
then so does for $\alpha$ along any subspace $W$ of $\mathbb{R}^{k}$ containing
$V$. Let $\alpha$ be a smooth $\mathbb{Z}^{k}$-action on a closed Riemannian
manifold $M$, $\mu$ an ergodic probability measure and $\Gamma$ the Oseledec
set. We show that, under a basic assumption on the Lyapunov spectrum, $\alpha$
has the shadowing property and is expansive on $\Gamma$ along any subspace $V$
of $\mathbb{R}^{k}$ containing a regular vector; furthermore, $\alpha$ has the
quasi-shadowing property on $\Gamma$ along any 1-dimensional subspace $V$ of
$\mathbb{R}^{k}$ containing a first-type singular vector. As an application, we
also consider the 1-dimensional subsystems (i.e., flows) with shadowing
property for the $\mathbb{R}^{k}$-action on the suspension manifold induced by
$\alpha$.

Dynamical Systems