  In this paper we consider those involutions $\theta$ of a finite-dimensional
Kac-Moody Lie superalgebra $\mathfrak g$, with associated decomposition
$\mathfrak g=\mathfrak k\oplus\mathfrak p$, for which a Cartan subspace
$\mathfrak a$ in $\mathfrak p_{\bar 0}$ is self-centralizing in $\mathfrak p$.
For such $\theta$ the restriction map $C_\theta$ from $\mathfrak p$ to
$\mathfrak a$ is injective on the algebra $P(\mathfrak p)^{\mathfrak k}$ of
$\mathfrak k$-invariant polynomials on $\mathfrak p$. There are five infinite
families and five exceptional cases of such involutions, and for each case we
explicitly determine the structure of $P(\mathfrak p)^{\mathfrak k}$ by giving
a complete set of generators for the image of $C_\theta$. We also determine
precisely when the restriction map $R_\theta$ from $P(\mathfrak g)^{\mathfrak
g}$ to $P(\mathfrak p)^{\mathfrak k}$ is surjective. Finally we introduce the
notion of a generalized restricted root system, and show that in the present
setting the $\mathfrak a$-roots $\Delta(\mathfrak a,\mathfrak g)$ always form
such a system.
