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A new bi-parametric $su(1,1)$ algebraization of the Heun class of equationsis 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 theGeneral Heun eqaution and its confluent versions. Explicit conditions leadingto these quasi-polynomials have been provided for the individual equations toallow direct use. For the Confluent and the Doubly-confluent Heun equations,specific parametric situations leading to (i) an infinite number ofquasi-polynomials and (ii) non-algebraizability of the equation have beenidentified.

Bi-parametric $su(1,1)$ structure of the Heun class of equations and quasi-polynomial solutions