We consider the differential system $$ \begin{cases} & y'(t)=a y(t)^3 + b z(t)\\ & z'(t)=c z(t)^5 - b y(t) \end{cases} $$ with $t>0$
$y(0)=y_0, z(0)=z_0,\quad a<0,\quad c<0,\quad b\in \mathbb{R}$
the question is to prouve that this system admits a unique solution on $[0,+\infty[$?
By standard ODE theorems, we know that if a solution doesn't exist, it will only not exist because the solution "blows up," that is, if there exists a $T>0$ such that $x(t)$ or $y(t)$ is unbounded as $t \to T^-$.
Multiply the first equation by $y(t)$, and the second solution by $z(t)$, and then add them together, you obtain $$ \frac12 \frac d{dt}((y(t))^2 + (z(t))^2) = a (y(t))^4 + c(z(t))^6 .$$ Since $a$ and $c$ are negative, it follows that $(y(t))^2 + (z(t))^2$ is bounded. Hence a blow up can never happen.