  In this paper we are concerned with higher regularity properties of the
elliptic system \[ \Delta\mathbf{u}=
|\mathbf{u}|^{q-1}\mathbf{u}\chi_{\{|\mathbf{u}|>0\}},\qquad\mathbf{u}=(u^1,\dots,u^m)
\] for $0\leq q<1$. We show analyticity of the regular part of the free
boundary $\partial\{|\mathbf{u}|>0\}$, analyticity of
$|\mathbf{u}|^{\frac{1-q}2} $ and $ \frac{\mathbf{u}}{|\mathbf{u}|}$ up to the
regular part of the free boundary. Applying a variant of the partial
hodograph-Legendre transformation and the implicit function theorem, we arrive
at a degenerate equation, which introduces substantial challenges to be dealt
with.
  Along the lines of our study, we also establish a Cauchy-Kowalevski type
statement to show the local existence of solution when the free boundary and
the restriction of $ \frac{\mathbf{u}}{|\mathbf{u}|} $ from both sides to the
free boundary are given as analytic data.
