  We study smooth integral curves of bidegree $(1,1)$, called \textit{smooth
conics}, in the flag threefold $\mathbb{F}$. The study is motivated by the fact
that the family of smooth conics contains the set of fibers of the twistor
projection $\mathbb{F}\to\mathbb{CP}^{2}$. We give a bound on the maximum
number of smooth conics contained in a smooth surface $S\subset\mathbb{F}$.
Then, we show qualitative properties of algebraic surfaces containing a
prescribed number of smooth conics. Lastly, we study surfaces containing
infinitely many twistor fibers. We show that the only smooth cases are surfaces
of bidegree $(1,1)$. Then, for any integer $a>1$, we exhibit a method to
construct an integral surface of bidegree $(a,a)$ containing infinitely many
twistor fibers.
