Modify locally the "array" environment for Equation Systems

You want to avoid the \tabcolsep altogether; note also the correction for the null delimiter. You can use an optional argument to enlarge the leading in special cases.

\documentclass{article}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage{amsmath}

\newenvironment{system}[1][1.1]
 {\renewcommand{\arraystretch}{#1}
  \left\{\begin{array}{@{}l@{}}}
 {\end{array}\right.\kern-\nulldelimiterspace}

\begin{document}
A centered formula for showing the center:
\[
\int_{-\infty}^{\infty}e^{-x^2}\,dx=\sqrt{\pi}
\]
The new \texttt{system} environment:
\[
\begin{system}
 x^{n}+y^{n}=z^{n}\\
 x^{m}+y^{m}=z^{m}
\end{system}
\]
Predefined \texttt{array} environment:
\[
\left\{%
 \begin{array}{l}
  x^{n}+y^{n}=z^{n}\\
  x^{m}+y^{m}=z^{m}
 \end{array}%
\right.
\]
A \texttt{system} environment with the optional argument:
\[
\begin{system}[1.5]
 x^{n}+y^{n}=z^{n}\\
 x^{m}+y^{m}=z^{m}
\end{system}
\]
\end{document}

The cases environment

This is one solution that after this time came to my mind, so I'll be able to use LyX with the cases environment defined as the more advanced dcases by re-defining cases using mathtools:

\documentclass{article}
%
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage{amsmath}
%
\usepackage{mathtools}
\renewenvironment{cases}{%
 \begin{dcases}%
}{%
 \end{dcases}%\kern-\nulldelimiterspace%
}
%
\begin{document}
A centered formula for showing the center:
\[
\int_{-\infty}^{\infty}e^{-x^2}\,dx=\sqrt{\pi}
\]
The new defined \texttt{cases} environment:
\[
\begin{cases}
 \int x^{n}+y^{n}=z^{n}\\
 x^{m}+y^{m}=z^{m}
\end{cases}
\]
Second method for \texttt{array} environment:
\[
\left\{%
 \begin{array}{l}
  \int x^{n}+y^{n}=z^{n}\\
  x^{m}+y^{m}=z^{m}
 \end{array}%
\right.
\]
%
\end{document}

The only thing that keeps me from inserting this lines inside my package LyXbasic is if this code does make some messing up with other math declarations.

So when inside LyX is typed this:

The output is as follows:

The difference in typesetting can be easily observed by a great improvement not only inside the pure LaTeX code but also "below" the command inside LyX.

The array environment

In order to modify correctly the array environment it has to be added this lines of code before the single environment:

\BeforeBeginEnvironment{array}{%
 \let\{\lbrace%
 \hskip-\arraycolsep%
}
\AfterEndEnvironment{array}{%
 \kern-\nulldelimiterspace%
} 

So if the single MWE is being added there will be a behaviour similar to @{}l using \hskip-\arraycolsep

\[
\left\{
 \begin{array}{l}
  x^n+y^n=z^n \\
  x^m+y^m=z^m 
 \end{array}
\right.
\]
\[
\begin{array}{ccc}
 a & b & c \\
 a & b & c \\
 a & b & c \\
 a & b & c 
\end{array}
\]

The output is as follows:

So now the common arraycolsep is not changed over the columns, but only aside the left delimiter.