= Solution
The <equilibrium points> are $(0,0)$ and $(\pm1,\pm(b-a))$, with correlated signs. The <Jacobian matrix> is $\begin{pmatrix}a-3bx^2&1\\3x^2-1&0\end{pmatrix}$. At the origin its eigenvalues satisfy $\lambda^2-a\lambda+1=0$. It is a <focus> for $0<|a|<2$, a repeated-eigenvalue node for $|a|=2$, and a <node> for $|a|>2$; stability is determined by the sign of $a$, with stability for $a<0$. Its <Poincaré index> is $+1$. At either other equilibrium the <determinant> is $-2$, so it is a <saddle point> with <Poincaré index> $-1$.
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