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

The solution $x_n\left(t\right)$, $n=1,2,$ of the $\mathit{\text{initial-values}}$ problem is reported of the $\mathit{\text{autonomous}}$ system of $2$ coupled first-order ODEs with $\mathit{\text{homogeneous}}\mathit{\text{cubic}}\mathit{\text{polynomial}}$ right-hand sides, \begin{eqnarray} \dot{x}n = c{n1} \left(x_1\right)^3 + c_{n2}\left( x_1\right)^2 x_2 + c_{n3} x_1 \left(x_2\right)^2+c_{n4} \left(x_2\right)^3
,\quad n=1,2\ , \nonumber \end{eqnarray} when the $8$ (time-independent) coefficients $c_{n\mathrm{\ell }}$ are appropriately defined in terms of $7$ $\mathit{\text{arbitrary}}$ 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 $8$ coefficients $c_{n\mathrm{\ell }},$ the $7$ parameters characterizing the solution of this model; and $2$ $\mathit{\text{constraints}}$ are $\mathit{\text{explicitly}}$ identified which, if satisfied by the $8$ parameters $c_{n\mathrm{\ell }},$ guarantee the $\mathit{\text{solvability}}\mathit{\text{by}}\mathit{\text{algebraic}}\mathit{\text{operations}}$ of this dynamical system. Also identified is a related, \textit{appropriately modified}, class of (generally $\mathit{\text{complex}}$) systems, reading \begin{eqnarray} \dot{{\tilde{x}}{n}} = $\mathbf{i}$\omega \tilde{x}{n} + c_{n1}\left(\tilde{x}{1}\right) ^{3}+c{n2}\left( \tilde{x}_{1}\right) ^2 \tilde{x}2 + c{n3}\tilde{x}_1 \left( \tilde{x}2\right)^2 + c{n4}\left(\tilde{x}_2 \right)^3\ ,\quad n=1,2\ , \nonumber \end{eqnarray} with $\mathbf{i}\omega$ an $\mathit{\text{arbitrary}}\mathit{\text{imaginary}}$ parameter, which feature the remarkable property to be $\mathit{\text{isochronous}}$, namely their $\mathit{\text{generic}}$ solutions are – as functions of $\mathit{\text{real}}\mathit{\text{time}}$ – $\mathit{\text{completely}}\mathit{\text{periodic}}$ with a period which is, for each of these models, a $\mathit{\text{fixed}}$ $\mathit{\text{integer}}\mathit{\text{multiple}}$ of the basic period $\tilde{T}=2\pi /\left\vert \omega \right\vert$.