  In this paper, the SUSY partner Hamiltonians of the quasi-exactly solvable
(QES) sextic potential $V^{\rm qes}(x) = \nu\, x^{6} + 2\, \nu\, \mu\,x^{4} +
\left[\mu^2-(4N+3)\nu \right]\, x^{2}$, $N \in \mathbb{Z}^+$, are revisited
from a Lie algebraic perspective. It is demonstrated that, in the variable $
\tau=x^2$, the underlying $\mathfrak{sl}_2(\mathbb{R})$ hidden algebra of
$V^{\rm qes}(x)$ is inherited by its SUSY partner potential $V_1(x)$ only for
$N=0$. At fixed $N>0$, the algebraic polynomial operator
$h(x,\,\partial_x;\,N)$ that governs the $N$ exact eigenpolynomial solutions of
$V_1$ is derived explicitly. These odd-parity solutions appear in the form of
zero modes. The potential $V_1$ can be represented as the sum of a polynomial
and rational parts. In particular, it is shown that the polynomial component is
given by $V^{\rm qes}$ with a different non-integer (cohomology) parameter
$N_1=N-\frac{3}{2}$. A confluent second-order SUSY transformation is also
implemented for a modified QES sextic potential possessing the energy
reflection symmetry. By taking $N$ as a continuous real constant and using the
Lagrange-mesh method, highly accurate values ($\sim 20$ s. d.) of the energy
$E_n=E_n(N)$ in the interval $N \in [-1,3]$ are calculated for the three lowest
states $n=0,1,2$ of the system. The critical value $N_c$ above which tunneling
effects (instanton-like terms) can occur is obtained as well. At $N=0$, the
non-algebraic sector of the spectrum of $V^{\rm qes}$ is described by means of
compact physically relevant trial functions. These solutions allow us to
determine the effects in accuracy when the first-order SUSY approach is applied
on the level of approximate eigenfunctions.
