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We introduce two new families of quasi-exactly solvable (QES) extensions ofthe oscillator in a $d$-dimensional constant-curvature space. For the firstthree members of each family, we obtain closed-form expressions of the energiesand wavefunctions for some allowed values of the potential parameters using theBethe ansatz method. We prove that the first member of each family has a hiddensl(2,$\mathbb{R}$) symmetry and is connected with a QES equation of the firstor second type, respectively. One-dimensional results are also derived from the$d$-dimensional ones with $d \ge 2$, thereby getting QES extensions of theMathews-Lakshmanan nonlinear oscillator.

Families of quasi-exactly solvable extensions of the quantum oscillator in curved spaces