Construction of unitals in Desarguesian planes

We present a new construction of non-classical unitals from a
classical unital $\cU$ in $\PG(2,q^2)$. The resulting
non-classical unitals are B--M unitals. The idea is to find a
non-standard model $\Pi$ of $\PG(2,q^2)$ with the following three
properties:
\begin{enumerate}[(i)]
\item points of $\Pi$ are those of $\PG(2,q^2)$;
\item
lines of $\Pi$ are certain lines and conics of $\PG(2,q^2)$;
\item
the points in $\cU$ form a non-classical B--M unital in $\Pi$.
\end{enumerate}
Our construction also works for the B--T unital, provided that conics are
replaced by certain algebraic curves of higher degree.