How many possibilities for eigenvectors are there for one eigenvalue? If I have a $2 \times 2$ matrix $A$, and then I find two eigenvalues $\lambda_1$ and $\lambda_2$ by subtracting $λI$ from $A$ and then taking the determinant=0(singular); to find $\lambda_1$ and $\lambda_2$. 
So for a one eigenvalue $\lambda_1$, how many possibilities are there for eigenvectors? in another words, how many solutions are there?   

I hope you are asking for maximum possiblity of independent eigen vector:  Below answer is for $2\times 2$ matrix.  
We know that if $\lambda_1\neq\lambda_2$ then corresponding eigen vector will be independent.  Below answer is based on this fact.
If $\lambda _1$ and $\lambda_2$ are different.... then there are only one independent eigen vector for corresponding eigen values.
If $\lambda_1$ and $\lambda_2$ are same then there may be two linear independent eigen vector.
