Can a harmonic vector field $X$ on a Riemannian surface $(M,g)$ possess a limit cycle(An isolated periodic orbit)?

Note that the Laplacian of a vector field is defined via natural correspondence between the space of vector fields and the space of $1$-forms.(The natural correspondence arising from the Riemannian metric). If this correspondence is denoted by $i$ then the Laplacian of a vector field $X$ is defined as $\Delta X=i^{-1} \Delta (i(X))$. Where the latter Laplacian is the natural Laplacian on the space of differential forms.

In particular is there a quadratic vector field $$\begin{cases} x'= ax+by+\lambda(x^2-y^2)+txy \\ y'=cx+dy+\mu (x^2-y^2)+sxy\end{cases}$$

which has at least one limit cycle?

The second question: Assume that $X$ is a vector field on a Riemannian surface Assume that $\gamma$ is a periodic orbit of $X$. Is it true to say that there is a point $p\in \gamma$ such that $\Delta X$ is tangent to $\gamma$ at $p$?