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