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.