2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/170224We introduce and study higher genera generalizations of the Halphen theory of continued fractions. The basic notion we start with is hyperelliptic Haplhen (HH) element $$\frac{\sqrt{X_{2g+2}}-\sqrt{Y_{2g+2}}}{x-y},$$ depending on parameter $y$, where $X_{2g+2}$ is a polynomial of degree $2g+2$ and $Y_{2g+2}=X_{2g+2}(y)$. We study regular and irregular HH elements. their continued fraction development and some basic properties of such development: even and odd symmetry and periodicity.29 pages, 4 figuresDynamical SystemsAlgebraic Geometry41A20; 14H05text