I mean it in a sense like in Morse theory (which is valid only for scalar-valued functions, as far as I know). More specifically, a critical point of a vector field is a point where the Jacobian has not full rank. For scalar functions degeneracy could be tested by checking the Hessian, but this is obviously not possible for vector fields.

The purpose is to find out, how a solution behaves in the neighborhood of a singularity, like annihilation/creation of solutions or a change in the dimension of the solution set.

I found the information that the Eigenvalues of the Jacobian are indicators for degeneracy (double Eigenvalues), but I can't link that definition of degeneracy to my application.

In the context of Morse theory a non degenerate critical point means that the Hessian is non degenerate. The Hessian is just the Jacobian of the gradient vector field, so in Morse theory a critical point is degenerate if and only if the gradient vector field has degenerate critical points in the sense that Robert Low is referring to. Note that the Jacobian is degenerate if and only one or more of the eigenvalues is zero. In particular if we have a family of vector fields parametrised by a parameter $t \in (-1, 1)$ say, the signal of a vector field becoming degenerate for $t = 0$ is that one (or more of) the eigenvalues tend to zero for $t = 0$.

The sense that you are using degenerate as two eigenvalues being the same is more subtle. If (at least) two eigenvalues are the same, the eigenvalue no longer determines the eigen spaces (over the complex numbers!) uniquely. Moreover we may not even be able to diagonalise the Jacobian even over the complex numbers (!) because there can be non trivial Jordan blocks (this does not happen for the Morse theory case, because the Hessian of a function in a critical point is automatically symmetric, and is therefore diagonalizable over the reals)

But what is the 'Hessian' of a vector-valued function? The Jacobian is already a matrix. Or let me ask this way: is there something like Morse theory applicable to vector functions?

By the way in the vector field on $R^2$ case

$$

F(x_1,x_2) = (F_1(x_1,x_2) , F_2(x_1,x_2),

$$

it is very easy to determine if two eigenvalues become the same: write down the characteristic polynomial of the Jacobian $J$ in the critical point (a, b).

$$

K(\lambda) = det(\lambda - J) = \lambda^2 - tr(J) \lambda + det(J)

$$

where the two by two matrix $J$ is given by

$$

J = (\frac{\partial F_i}{\partial x_j})|_{(x_1,x_2) = (a, b)} ( i = 1, 2; j = 1,2)

$$

The eigenvalues are the solutions of $K(\lamda) = 0$, so the Jacobian has two equal eigenvalues if and only if the discriminant is zero, i.e. if:

$$

tr(J)^2 - 4det(J) = 0.

$$