Systeme package: having the "constants" on the left while maintaining package benefits?

Question

Edit: Upon further reflection, the problem stated below could be resolved if two systeme tables could be placed side by side while maintaining vertical alignment of rows. Is it possible -- again in a reasonable way -- to have the rows of the following two tables aligned?

\documentclass{article}
\usepackage{systeme}
\usepackage{amsmath}
\begin{document}

\syslineskipcoeff{2.5}
\systeme{\dfrac{dx}{dt}\Relbar,\dfrac{dy}{dt}\Relbar}\sysdelim..\systeme{y+x^2,\dfrac{1}{2}x}

\end{document}

---

Original question below:

Using the systeme package, equations are entered as

\[\systeme{x_1+x_3=y_1,x_2+x_1+x_3=y_2}\]

with the variables on the left, and the "constants" on the right. If entered in this way, the package will automatically sort the variables and introduce spacing for missing variables. Is it possible, in a reasonably straightforward way, to either

1. Have the package apply the same formatting to the expressions on either side of the equals sign, not just the left hand side.
2. Have the package apply the formatting that it currently does to the left hand side, to the right hand side only.

I suspect this isn't easy but thought I'd ask anyway.

MWE:

\documentclass{article}
\usepackage{systeme}
\begin{document}

\[\systeme{x_1+x_3=y_1,x_2+x_1+x_3=y_2}\]

Auto sorting with spacing.

\[\systeme{y_1=x_1+x_3,y_2=x_2+x_1+x_3}\]

Unsorted, without spacing.

\end{document}