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We consider two families of extensions of the oscillator in a $d$-dimensionalconstant-curvature space and analyze them in a deformed supersymmetricframework, wherein the starting oscillator is known to exhibit a deformed shapeinvariance property. We show that the first two members of each extensionfamily are also endowed with such a property provided some constraintconditions relating the potential parameters are satisfied, in other words theyare conditionally deformed shape invariant. Since, in the second step of theconstruction of a partner potential hierarchy, the constraint conditionschange, we impose compatibility conditions between the two sets to buildpotentials with known ground and first excited states. To extend such resultsto any members of the two families, we devise a general method wherein thefirst two superpotentials, the first two partner potentials, and the first twoeigenstates of the starting potential are built from some generating function$W_+(r)$ (and its accompanying function $W_-(r)$).

Deformed shape invariance symmetry and potentials in curved space with two known eigenstates