Let $C$ be a family of smooth (all but one curve is smooth, I'll explain later) cubic curves in $\mathbb{P}_{\mathbb{C}}^2$ with the following property: given any two elements $a,b \in C$, the curves $a$ and $b$ intersect only at one point, namely $p$, with intersection multiplicity $9$. Because they only intersect once, every point in the projective plane other than $p$ lies on at most one cubic curve in $C$. Furthermore, we've shown that each point in the projective plane other than $p$ lies on exactly one cubic curve in $C$. Hence, we can assign to each point in $\mathbb{P}_{\mathbb{C}}^2$, a tangent vector. Thus we can construct a vector field on the projective plane this way.
Concerning the one non-smooth cubic curve: It has a nodal singularity at $p$. So when we construct the vector field, we can simply take the tangent vector that corresponds with all of the other cubic curves.
I'm wondering if such a vector field that arises from a situation like this has a name, or whether it is even an interesting object to look at.
Questions I'm wondering about: Which vector fields can be constructed in this way? Do all of the vector fields constructed in this way share certain properties? etc. I'm an undergraduate looking to write a senior thesis this year and this would be an extension of my REU project from the summer.
Probably implicit differentiation and gradient flows, for functions on the plane. Your families might be projectivizations of $F(x,y)=c$ with $c$ variable and $F$ fixed. Implicit differentiation gives the vector field perpendicular to yours, and gradient flow of $F$ might be the thing you have in mind, up to a choice of parameterization, or a replacement of $F$ by another function with the same level sets.