Solve a system of non linear equations over GF I have the following set of equations: 

$$M_{1}=\frac{y_1-y_0}{x_1-x_0}$$

$$M_{2}=\frac{y_2-y_0}{x_2-x_0}$$

$M_1, M_2,  x_1, y_1, x_2, y_2,$ are known and they are chosen from a $GF(2^m)$. I want to find  $x_0,y_0$

Does the previous set of equations is solvable?

And more...

If I have the following set of equations:

$$M_1=\frac{k_1-(y_0+(\frac{y_1-y_0}{x_1-x_0})(l_1-x_0))}{(l_1-x_0)(l_1-x_1)}$$

$$M_2=\frac{k_2-(y_0+(\frac{y_1-y_0}{x_1-x_0})(l_2-x_0))}{(l_2-x_0)(l_2-x_1)}$$

$$M_3=\frac{k_3-(y_0+(\frac{y_1-y_0}{x_1-x_0})(l_3-x_0))}{(l_3-x_0)(l_3-x_1)}$$

$$M_4=\frac{k_4-(y_0+(\frac{y_1-y_0}{x_1-x_0})(l_4-x_0))}{(l_4-x_0)(l_4-x_1)}$$

where $x_0,y_0 x_1,y_1$ are the unknown GF elements.

As Dilip Sarwate clarified the set of equations is constructed by someone who " chose three distinct x0,x1,x2, as well as y0,y1,y2, then computed M1, M2, and finally revealed $M_1,M_2,x_1,y_1,x_2,y_2$ but not $x_0,y_0$ to us" i.e. it is known that the system has solution.

My question was: 
Can I recover the $x_0, y_0$ or in the second set of equations can I recover $x_0, x_1, y_0, y_1$ and generally in nonlinear sets to recover the respective $x_i, y_i$  by the provided info, on a GF? and the main point of my question: Does the fact that the set of equations is defined on a Galois Field impose any difficulties to find its solution? 

I was not sure that it is possible to compute the solution of the  problem with the aforementioned parameters on a Galois Field.  

As Dilip Sarwate stated in his answer the solution of the previous problem can be recovered for linear and nonlinear equations.
There have been extensive comments by the OP on this question as well as a related one and its answers and the consensus
don't seem to be converging at all to anything sensible.


  Broadly speaking, the field in which we are operating influences
  the answer to the question of whether a system of equations has
  solutions or not to some extent, but not in the ways that the OP
  thinks it does. Whether we are operating in a prime field or an
  extension of a prime field (what the OP calls gf or GF) has relatively
  little to do with the matter. In particular, for linear equations,
  the general theory of linear equations over a field
  usually has more to say about the matter than the identity of the
  field.


As Henrick Hellström has pointed out in a comment on the question,
the first set of equations in the OP's question can be converted
into a pair of linear equations in the unknowns $x_0$ and $y_0$,
say
$$\begin{align} a_{11}x_0 + a_{12}y_0 &amp;= b_1\\
a_{21}x_0 + a_{22}y_0 &amp;= b_2\end{align}$$
where the $a_{ij}$ and the $b_k$ are functions of
the known quantities $M_1, M_2, x_1, y_1, x_2, y_2$.
I refuse to calculate and state explicitly what the $a_{ij}$ 
and the $b_k$ are in terms of the known quantities. I will
differ from Henrick slightly, though, in that I do not think
that these equations are solvable for each and every choice of
$M_1, M_2, x_1, y_1, x_2, y_2$. This has nothing to do with
whether these quantities belong to GF$(2^m)$ or GF$(p^m)$ for
$p &gt; 2$ or GF$(p)$ and everything to do with basic linear
equation theory: the matrix might be singular
for some choices of $M_1, M_2, x_1, y_1, x_2, y_2$ in which
case we get multiple solutions rather than a unique solution,
or the matrix might be singular and the equations might be
 inconsistent in which case no solution exists.

All this, however, as well as the OP's questions about other
equations is irrelevant in the context of Shamir's secret
sharing scheme which is referenced in the 
other question
but not in this one but which I believe is the reason for
these questions.

If someone chose three distinct $x_0, x_1, x_2$, as well
as $y_0, y_1, y_2$, then computed $M_1$, $M_2$, and finally
revealed $M_1, M_2, x_1, y_1, x_2, y_2$, but not $x_0, y_0$
to us, then we can find $x_0$ and $y_0$ from the given
relationships, and inconsistency is not an issue.  Note the
requirement that $x_0, x_1, x_2$ be distinct which allows
the unnamed someone to avoid the heartbreak of
division by $0$ when computing $M_1$ and $M_2$.

Similar remarks apply to the nonlinear equations in the
OP's question as well.