  In this article, we investigate the regularity for certain elliptic systems
without a $L^2$-antisymmetric structure. As applications, we prove some
$\epsilon$-regularity theorems for weakly harmonic maps from the unit ball $B=
B(m) \subset \mathbb{R}^m $ $(m\geq2)$ into certain pseudo-Riemannian
manifolds: standard stationary Lorentzian manifolds, pseudospheres
$\mathbb{S}^n_\nu \subset \mathbb{R}^{n+1}_\nu$ $(1\leq\nu \leq n)$ and
pseudohyperbolic spaces $\mathbb{H}^n_\nu \subset \mathbb{R}^{n+1}_{\nu+1}$
$(0\leq\nu \leq n-1)$. Consequently, such maps are shown to be H\"{o}lder
continuous (and as smooth as the regularity of the targets permits) in
dimension $m=2$. In particular, we prove that any weakly harmonic map from a
disc into the De-Sitter space $\mathbb{S}^n_1$ or the Anti-de-Sitter space
$\mathbb{H}^n_1$ is smooth. Also, we give an alternative proof of the
H\"{o}lder continuity of any weakly harmonic map from a disc into the
Hyperbolic space $\mathbb{H}^n$ without using the fact that the target is
nonpositively curved. Moreover, we extend the notion of generalized (weakly)
harmonic maps from a disc into the standard sphere $\mathbb{S}^n$ to the case
that the target is $\mathbb{S}^n_\nu$ $(1\leq\nu \leq n)$ or $\mathbb{H}^n_\nu$
$(0\leq\nu \leq n-1)$, and obtain some $\epsilon$-regularity results for such
generalized (weakly) harmonic maps.
