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We identify many new solvable subcases of the general dynamical systemcharacterized by two autonomous first-order ordinary differential equationswith purely quadratic right-hand sides; the solvable character of thesedynamical systems amounting to the possibility to obtain the solution of theirinitial value problem via algebraic operations. Equivalently---by consideringthe analytic continuation of these systems to complex time---theiralgebraically solvable character corresponds to the fact that their generalsolution is either singlevalued or features only a finite number of algebraicbranch points as functions of complex time (the independent variable). Thus ourresults provide a major enlargement of the class of solvable systems beyondthose with singlevalued general solution identified by Garnier about 60 yearsago. An interesting property of several of these new dynamical systems is theelementary character of their general solution, identifiable as the roots of apolynomial with explicitly obtainable time-dependent coefficients. We alsomention that, via a well-known time-dependent change of (dependent andindependent) variables featuring the imaginary parameter $% \mathbf{i} \omega $(with $\omega $ an arbitrary strictly positive real number), autonomousvariants can be explicitly exhibited of each of the algebraically solvablemodels we identify: variants which all feature the remarkable property to beisochronous, i.e. their generic solution is periodic with a period that is afixed integer multiple of the basic period $T=2\pi/\omega$.

New algebraically solvable systems of two autonomous first-order ordinary differential equations with purely quadratic right-hand sides