Solvable systems of two coupled first-order ODEs with homogeneous cubic polynomial right-hand sides

The solution xnt, n = 1, 2, of the initial-value problem is reported for the autonomous system of two coupled first-order ordinary differential equations with homogeneous cubic polynomial right-hand sides, ẋn=cn1x13+cn2x12x2+cn3x1x22+cn4x23,n=1,2, when the eight (time-independent) coefficients cnℓ are appropriately defined in terms of seven arbitrary parameters, which then also identify the solution of this model. The inversion of these relations is also investigated, namely, how to obtain, in terms of the eight coefficients cnℓ, the seven parameters characterizing the solution of this model, and two constraints are explicitly identified, which, if satisfied by the eight parameters cnℓ, guarantee the solvability by algebraic operations of this dynamical system. Also identified is a related, appropriately modified, class of (generally complex) systems, reading x̃ṅ=iωx̃n+cn1x̃13+cn2x̃12x̃2+cn3x̃1x̃22+cn4x̃23,n=1,2, with iω being an arbitrary imaginary parameter, which features the remarkable property to be isochronous, namely, their generic solutions are—as functions of real time—completely periodic with a period that is, for each of these models, a fixed integer multiple of the basic period T̃=2π/ω.