  Let $X$ be a compact Riemann surface of genus $g \geq 3$. Let $\cat{M}_{Hod}$
denote the moduli space of stable $\lambda$-connections over $X $ and
$\cat{M}'_{Hod} \subset \cat{M}_{Hod}$ denote the subvariety whose underlying
vector bundle is stable. Fix a line bundle $L$ of degree zero. Let
$\cat{M}_{Hod}(L)$ denote the moduli space of stable $\lambda$-connections with
fixed determinant $L$ and $\cat{M}'_{Hod}(L) \subset \cat{M}_{Hod}(L)$ be the
subvariety whose underlying vector bundle is stable. We show that there is a
natural compactification of $\cat{M}'_{Hod}$ and $\cat{M}'_{Hod} (L)$, and
study their Picard groups. Let $\M_{Hod}(L)$ denote the moduli space of
polystable $\lambda$-connections. We investigate the nature of algebraic
functions on $\cat{M}_{Hod}(L)$ and $\M_{Hod}(L)$. We also study the
automorphism group of $\cat{M}'_{Hod}(L)$.
