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