on with external focusing by linear fields:</p>
<div class="math notranslate nohighlight">
\[\displaystyle r '' + \frac {1} {\beta ^ 2 \gamma} \gamma 'r' + \frac {1} {2 \beta ^ 2 \gamma} \gamma''r + kr - \frac { 2I} {I_a (\beta \gamma) ^ 3} \frac {1} {r} - \frac {\epsilon ^ 2} {r ^ 3} = 0;\]</div>
<p>and for an elliptical beam:</p>
<div class="math notranslate nohighlight">
\[\begin{split}\begin{equation*}
 \begin{cases}
   \displaystyle a '' + \frac {1} {\beta ^ 2 \gamma} \gamma 'a' + \frac {1} {2 \bet