Solving linear system, finding equilibrium and bifurcation points For homework, I have to solve the following problem: consider the system of ODE's
\begin{equation} 
x'=-x^2+a \\ y'=-y
\end{equation}
where the parameter $a$ is a real number. I have to characterize the equilibrium points and draw a bifurcation diagram. I am asked to linearize the system first. I have missed a couple of classes and I am struggling to understand how I am supposed to work here. 
If $a$ is positive, then there are two equilibrium points $(\sqrt{a},0)$ and $(-\sqrt{a},0)$, if $a$ is equal to zero, then there is one equilibrium point $ \left ( 0,0 \right )$ and finally  if $a$ is negative,  then no equilibrium points exist. The Jacobi matrix of the system is 
\begin{pmatrix}
 - 2x & 0\\ 
0 & -1
\end{pmatrix}
which is diagonal, hence its eigenvalues are $-2x$ and $-1$. Then, I calculate this matrix at the equilibrium points:
If $a>0$, then the two Jacobi matrices are
\begin{pmatrix}
-2\sqrt{a}  & 0\\ 
0 & -1
\end{pmatrix}
which has two real and negative eigenvalues, so $(\sqrt{a},0)$ is a stable point and 
\begin{pmatrix}
2\sqrt{a}  & 0\\ 
0 & -1
\end{pmatrix}
which has one negative and one positive eigenvalue, so  $(-\sqrt{a},0)$ is a saddle point.
If $a=0$ the Jacobi matrix is
\begin{pmatrix}
0 & 0\\ 
0 & -1
\end{pmatrix}
which has one zero eigenvalue so the linearization method does not provide any information about the stability of $\left ( 0,0 \right )$. As my notes suggest, for this point we find the eigenvectors of this matrix and the orbits of the solutions. But I do not understand why and what information we obtain from this. What can I say about the stability of $\left ( 0,0 \right )$ then? What about the saddle point, do I use a similar method? I understand that there is a bifurcation for $a=0$ but I don't know how am I supposed to draw the diagram without a computer. Another thing I find in the notes in another example is that we divide the equations and find a relationship between the solutions which does not involve time, for some reason. 
Thanks in advance, any insight on this will be appreciated, right now I am lost. 

Hint:
By integration, when $a=0$, either
$$x=0$$ or
$$\frac1x=t+C.$$
