  We consider two families of extensions of the oscillator in a $d$-dimensional
constant-curvature space and analyze them in a deformed supersymmetric
framework, wherein the starting oscillator is known to exhibit a deformed shape
invariance property. We show that the first two members of each extension
family are also endowed with such a property provided some constraint
conditions relating the potential parameters are satisfied, in other words they
are conditionally deformed shape invariant. Since, in the second step of the
construction of a partner potential hierarchy, the constraint conditions
change, we impose compatibility conditions between the two sets to build
potentials with known ground and first excited states. To extend such results
to any members of the two families, we devise a general method wherein the
first two superpotentials, the first two partner potentials, and the first two
eigenstates of the starting potential are built from some generating function
$W_+(r)$ (and its accompanying function $W_-(r)$).
