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 $\gamma: I\subset \mathbb{R} \rightarrow \mathbb{R}^2$.
My question is the following: Is it possible to always find coefficients $a_{ij}\in \mathbb{R}$ such that $$\sum_{i,j=1}^{\infty}a_{ij}\cdot \gamma_{i}(t)^i \cdot \gamma_{i}(t)^j=0 \quad \forall t \in I$$ Or to reframe the question, for a given trajectory of a polynomial vector field, can we always find an analytic function which has the solution trajectory as zero set?