Let's assume we have a two dimensional polynomial vector field of degree $d$
$$F: \mathbb{R}^{2}\rightarrow\mathbb{R}^{2}, \quad (x,y)\mapsto \begin{pmatrix}P(x,y), \\ Q(x,y)\end{pmatrix}$$
and we are given a solution $\gamma: I\subset \mathbb{R} \rightarrow \mathbb{R}^2$.
Let's define a polynomial of the same degree $d$ $$G:\mathbb{R}^{2}\rightarrow\mathbb{R}, \quad (x,y)\mapsto G(x,y).$$ but which can have completely different coefficients and also other monomials. I.e. $x^5 \cdot y$ could be a monomial in $G$ which is not a monomial of $F$.
Do we know if there exists an initial condition $x_0$ such that the solution curve $\gamma(t,x_0)$ fulfills
$$G(\gamma_x(t,x_0),\gamma_y(t,x_0)=0 \quad \forall t?$$
As an example, given $F=(-y,x)$ and $x_0=(1,0)$, the solution fulfills $$G(x,y)=x^2+y^2-1=0$$ But $G$ is a degree higher than $F$. But that is the only example I currently was able to come up.
I am interested in any result and any new example of this kind.
What you're looking for is the definition of a Darboux polynomial. A first integral for your vector field $F$ is a function $f(x,y)$ such that $$\nabla f \cdot F=0,$$ that is: all curves $f(x,y)=C$ (any $C$ real) are integral curves for $F$.
A Darboux polynomial (or second integral) is a curve $g(x,y)=0"$ such that $$\nabla g \cdot F=hg$$ for some polynomial $h(x,y)$. In this way we have $\nabla g \cdot F=0$ along the curve $g=0$, so that curve is a particular integral curve of $F$.
As an example, consider $F=(-2x^2-2xy-6y^2-6y+2,3x^2+4xy+6x+y^2-1)$, you can easily check that, for $g(x,y)=x^2+y^2-1$, we have $\nabla g \cdot F=hg$ with $h(x,y)=2y-4x$. Below we have plotted the vector field $F$, some of it's integral curves (in red) and two Darboux curves (in blue and green)
Note that both the vector field and the particular integral curve are of degree two.
In fact you can produce as much examples as you want, just take any antisymmetric matrix $S$, two curves $g(x,y)=0$ and $f(x,y)=0$ and a vector field $F=S \nabla (gf)$. In this way both $g(x,y)=0$ and $f(x,y)=0$ will be integral curves.
An nice reference with a ton of examples and pictures is Goriely's Integrability and Nonintegrability of Dynamical Systems.