Finite field special functions I am looking for special class of involutive functions (f) with exactly one (zero) fixed point over finite field $F_q$ with properties:
1) $f(f(x))=x$ for any $x \in F_q$ , $f(x)=x$ iff $x=0$
2)   For any fixed $a \in F_q$, $a \ne 0$ the following holds:
$f(x+a)-f(x)=g_a(x)$
where $g_a(x_1) \ne g_a(x_2) $ for any $x_1 \ne x_2$,
$x_1 \ne -a$ and $x_1 \ne 0$, $x_2 \ne -a$ and $x_2 \ne 0$
and
for any $x \ne -a$ and $x \ne 0$
$g_a(x) \ne a$
EDIT(25 Aug 2019) I'd like to add 3'rd constraint:
3)   For any fixed $a \in F_q$, $a \ne 0$ the following holds:
$f(x+a)-x=g_a(x)$
where $g_a(x_1) \ne g_a(x_2) $ for any $x_1 \ne x_2$,
$x_1 \ne -a$ and $x_1 \ne 0$, $x_2 \ne -a$ and $x_2 \ne 0$
 A: What you are looking for are called perfect nonlinear or differentially $1-$uniform functions.
They don't exist over even characteristic since if $x_0$ satisfies
$$
f(x+a)-f(x)=b,
$$
so does $x_0+a.$
For a long time only some power functions or functions equivalent to them were known. A recent paper lists
the following known examples among others.
$$x^2~~ in ~~GF(p^n), $$
$$x^{p^k+1} ~~in ~~GF(p^n),\quad k \leq n/2~~and ~~n/(k,n)~~odd$$
$$x^{10} + x^6 − x^2 ~~in~~ GF(3^n), ~~n \geq 5 ~~odd$$
See New families of perfect nonlinear polynomial functions by
Zhengbang Zha, Xueli Wang, Journal of Algebra (322):3912-3918.
for more. One class of such polynomials are called Dembowski-Ostrom polynomials. All of the above are, but a recent example $$x^{(3^k+1 )/2}$$ isn't.
Edit: I apologise, I shouldn't post before my morning coffee. This is now essentially a long comment.
The functions displayed are not involutions. Recent work on involutions is here. I think your question is quite difficult.
A: Taking into account Peter's Tailor analysis I can say the following.
For example we have finite field $GF(p)$ for $p$ prime;
We can take element $2$ and obtain subgroup generated by $2$: $<2>$.
This subgroup generates set of cosets:
$<2>$, $C_1<2>$ .... $C_{K-1}<2>$
where $k=(|GF^*(p)|/(|<2>|))$.
So exponential functions means that 
f($C_i<2>$) = $C_j<2>$
and we have involution of cosets (taking into account that their number is even)
$(i_1,j_1),...,(i_r,j_r)$.
Plus we have the following condition:
for any two cosets $i,j$
if $(a,b)\in C_i<2>$, $(f(a),f(b))\in C_j<2>$
we have: $a/b$=$f(a)/f(b)$.
So we can iterate over all involutions on the set of such Cosets
and iterate over corresponding mapping for each pair of cosets.
A: The perfect nonlinear function $f$ can be represented as a special sequence ($a_{1},...,a_{n}$).
The main property: $f(a_{i}) = a_{-i}$.
Other properties:
${a_{2^k}}$ = $2^k$;
There exist such element $\alpha$ that
${a_{-i} = a_{i} + i*(\alpha-1)}$
For $i \ne 0, i+1 \ne 0$ we have:
$ f(a_{i}) - f(a_{i}-a_{i+1}) = 1$
For any 2 not equal $i,k \ne 0, i+k \ne 0 $:
$ f(a_{i}) - f(a_{i}-a_{i+k}) = a_{k}$
I know yet 5 of the same nature sequences for the $GF(31)$.
I know a sequence of the same nature for the $GF(2^{k}-1)$ and prime $2^{k}-1$.
Looks like the following property holds as well:
$
a_{j} - a_{k} = a_{j+ \frac{a_{k}-a_{k-j}}{\alpha-1}}
$
$
 a_{i} = -f(i*(\alpha-1))
$
Example ($\alpha=18$):

|index   |  value |
|--------| -------|
|0       |   0    | 
|1       |   1    |
|2       |   2    | 
|3       |   23   |
|4       |   4    |
|5       |   19   |
|6       |   15   |
|7       |   3    |
|8       |   8    |
|9       |   28   |
|10      |   7    |
|11      |   13   |
|12      |   30   |
|13      |   21   |
|14      |   6    |
|15      |   9    |
|16      |   16   |
|17      |   27   |
|18      |   25   |
|19      |   17   |
|20      |   14   |
|21      |   22   |
|22      |   26   |
|23      |   20   |
|24      |   29   |
|25      |   24   |
|26      |   11   |
|27      |   10   |
|28      |   12   |
|29      |   5    |
|30      |   18   |
--------------------


