In this question, we work over an unspecified algebraically closed field, not necessarily of characteristic zero.
Let $\mathscr F$ be the $1$-dimensional foliation of $\mathbb P^3$ defined by the polynomial vector field
$$\mathscr F : P \frac \partial {\partial x} + Q \frac \partial {\partial y} + R \frac \partial {\partial z}$$
away from the plane at infinity $L \cong \mathbb P^2$. Assuming $L$ is not $\mathscr F$-invariant, which is the case for a generically chosen plane, then $P, Q, R$ are of the form
$$P = xC + F, \qquad Q = yC + G, \qquad R = zC + H,$$
where $C$ is a homogeneous polynomial of degree $d$, and $F, G, H$ are arbitrary polynomials of degree $\le d$. The number $d \in \mathbb N$ is called the degree of the foliation. Geometrically, $C$ is the locus of points where $\mathscr F$ is tangent to $L$.
For reasons that can be seen in my previous question, I would like $C$ to be a smooth curve. The foliation $\mathscr F$ is fixed, but I am allowed to perform a linear change of coordinates of $\mathbb P^3$, which basically amounts to selecting a different plane at infinity. Can I always find a plane $L \subset \mathbb P^2$ that is not $\mathscr F$-invariant, and such that the curve $C$ where $\mathscr F$ is tangent to $L$ is smooth?