  A new bi-parametric $su(1,1)$ algebraization of the Heun class of equations
is explored. This yields additional quasi-polynomial solutions of the form
$\{z^{\alpha}P_N(z): \ \alpha \in \mathbb{C}, \ N \in \mathbb{N}_0\}$ to the
General Heun eqaution and its confluent versions. Explicit conditions leading
to these quasi-polynomials have been provided for the individual equations to
allow direct use. For the Confluent and the Doubly-confluent Heun equations,
specific parametric situations leading to (i) an infinite number of
quasi-polynomials and (ii) non-algebraizability of the equation have been
identified.
