A saddle-node bifurcation for a vector field on a plane $\dot{x} = f(x, \mu)$, where $x \in \mathbb{R}^{2}$ and $\mu \in \mathbb{R}$ corresponds to a creation of two equilibria (both hyperbolic, one saddle, the other node) out of zero, as the parameter $\mu$ passes some critical value.
Is it possible to have a scenario where two hyperbolic equilibria, are born such that $ \textbf{both} $ are nodes? (Say both would be (un)stable, or one stable, the other unstable). ( I call this hypothetical scenario node-node bifurcation).
This leads to a more general question: is it possible for a vector field on a plane to have $\textbf{only 2 fixed points such that both are nodes}$, and no other fixed points?
I've been unable to come up with an example of such a vector field, or come up with a reason why this is not possible.
Edit to simplify this example:
Let $H(x,y,\lambda) = \left(\lambda - x^2,-x\cdot y\right)$. Here is a phase portrait for $H(\cdot, \cdot,1)$.
As $\lambda \to 0$, the two nodes move towards each other and annihilate each other at $\lambda = 0$.