I am very new to Noether's theorem and in our (first!) mechanics class it was proved using generators $X_i$,$X$ of a Lie group. Because I didn't really understand this proof I have trouble solving the following problem:
A particle of mass $m=1$ moves through a potential $V(\vec{r})$ which is invariant under the transformation
$$\left(\begin{array}{c}x \\ y \\ z \end{array}\right) \mapsto \left(\begin{array}{c} x\cos\varphi-y\sin\varphi\\ x\sin\varphi + y\cos\varphi\\ z+c\varphi\end{array}\right)$$
Now I have to calculate the conserved Noether current
$$J=\frac{\partial L}{\partial\dot{q}_i}X_i+ \left(L-\frac{\partial L}{\partial\dot{q}_i}\dot{q}_i\right)X.$$
My problem is finding the generators $X$ and $X_i$.
Every advice, hint, tip, etc. is very much appreciated!
$\textbf{Edit:}$ The action is thought to be invariant under the 1-parameter group $t\rightarrow t'=g(t,s)$, $q_i\rightarrow q_i'=g(q_i,s)$ where the depency on $s$ is continuous.
Infinitessimal transformations are defined as $t'=t+\delta t$ with $\delta t = X\delta s$ and $q_i'=q_i+\delta q_i$ with $\delta q_i=X_i\delta s$.
$\textbf{Edit 2:}$ Building on the answer of lux the transformation in matrix form looks like $$\left(\begin{array}{ccc}\cos\varphi&&-\sin\varphi&&0\\ \sin\varphi&&\cos\varphi&&0\\ 0&&0&&1+\frac{c\varphi}{z}\end{array}\right).$$ Differentiating this with respect to $\varphi$ gives $$A(\varphi)=\left(\begin{array}{ccc} -\sin\varphi&&-\cos\varphi&&0\\ \cos\varphi&&-\sin\varphi&&0\\0&&0&&\frac{c}{z} \end{array}\right).$$ Then
$$A(0)=\left(\begin{array}{ccc}0&&-1&&0\\ 1&&0&&0\\0&&0&&\frac{c}{z}\end{array}\right).$$
(The following is just speculation which is inspired from the example of the rotation around the $z$-axis which I looked at to know how to continue in this case.)
The generator would the be $X_i =A(0)\cdot\vec{x}$, where $\vec{x}=(x,y,z)^T)$.
Then using the above expression for the Noether current $J$ I get that
$$\frac{\partial L}{\partial \dot{q}_i}X_i=L_z+p_z\cdot c,$$ where $L_z$ is the $z$-component of the angular momentum and $p_z$ is the $z$-component of the momentum. This is based on the matrix $A$ for which I am not very sure about the entry $a_{33}$...