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We apply Kovacic's algorithm from differential Galois theory to show that allcomplex non-oscillatory solutions (finite exponential of convergence of zeros)of certain Hill equations considered by Bank and Laine using Nevanlinna theorymust be Liouvillian solutions. That is, solutions are obtainable by suitabledifferential field extensions construction. In particular, we have establisheda full correspondence between solutions of non-oscillatory type and Liouvilliansolutions for a particular Hill equation. Explicit closed-form solutions areobtained via both methods for this Hill equation whose potential has fourexponential functions in the Bank-Laine theory. The differential equation is aperiodic form of biconfluent Heun equation. We further show that theseLiouvillian solutions exhibit novel single and double orthogonality and aFredholm integral equation over suitable integration regions in $\mathbf{C}$that mimic single/double orthogonality for the corresponding Liouvilliansolutions of the Lam\'e and Whittaker-Hill equations, discovered by Whittakerand Ince almost a century ago.

Galoisian approach to complex oscillation theory of some Hill equations