  The main objective of this paper is to derive the Enneper-Weierstrass
representation of minimal surfaces in $\mathbb{E}^3$ using the soliton surface
approach. We exploit the Bryant-type representation of conformally parametrized
surfaces in the hyperbolic space $H^3(\lambda)$ of curvature $-\lambda^2$,
which can be interpreted as a 2 by 2 linear problem involving the spectral
parameter $\lambda$. In the particular case of constant mean
curvature-$\lambda$ surfaces a special limiting procedure $(\lambda\rightarrow
0)$, different from that of Umehara and Yamada [33], allows us to recover the
Enneper-Weierstrass representation. Applying such a limiting procedure to the
previously known cases, we obtain Sym-type formulas. Finally we exploit the
relation between the Bryant representation of constant mean curvature-$\lambda$
surfaces and second-order linear ordinary differential equations. We illustrate
this approach by the example of the error function equation.
