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Motivated by the wild behavior of isolated essential singularities in complexanalysis, we study singular complex analytic vector fields $X$ on arbitraryRiemann surfaces $M$. By vector field singularities we understand zeros, poles,isolated essential singularities and accumulation points of the above kind.  In this framework, a singular analytic vector field $X$ has canonicallyassociated; a 1-form, a quadratic differential, a flat metric (with a geodesicfoliation), a global distinguished parameter or $\mathbb{C}$-flow box $\Psi_X$,a Newton map $\Phi_X$, and a Riemann surface $\mathcal{R}_X$ arising from themaximal $\mathbb{C}$-flow of $X$.  We show that every singular complex analytic vector field $X$ on a Riemannsurface is in fact both a global pullback of the constant vector field under$\Psi_X$ and of the radial vector field on the sphere under $\Phi_X$.  As a result of independent interest, we show that the maximal analyticcontinuation of the a local $\mathbb{C}$-flow of $X$ is univalued on theRiemann surface $\mathcal{R}_{X} \subset M \times \mathbb{C}_t$, where$\mathcal{R}_X$ is the graph of $\Psi_{X}$.  Furthermore we explore the geometry of singular complex analytic vectorfields and present a geometrical method that enables us to obtain the solution,without numerical integration, to the differential equation that provides the$\mathbb{C}$-flow of the vector field.  We discuss the theory behind the method, its implementation, comparison withsome integration-based techniques, as well as examples of the visualization ofcomplex vector fields on the plane, sphere and torus.  Applications to visualization of complex valued functions is discussedincluding some advantages between other methods.

On the geometry, flows and visualization of singular complex analytic vector fields on Riemann surfaces