  Amplitudes for the reaction $\pi^-p\to \Lambda K^0$ are reconstructed from
data on the differential cross section $d\sigma/d\Omega$, the recoil
polarization $P$, and on the spin rotation parameter $\beta$. At low energies,
no data on $\beta$ exist, resulting in ambiguities. An approximation using $S$
and $P$ waves leads only to a fair description of the data on $d\sigma/d\Omega$
and $P$; in this case, there are two sets of amplitudes. Including $D$ waves,
the data on $d\sigma/d\Omega$ and $P$ are well reproduced by the fit but now,
there are several distinct solutions which describe the data with identical
precision. In the range where the spin rotation parameter $\beta$ was measured,
a full and unambiguous reconstruction of the partial wave amplitudes is
possible. The energy-independent amplitudes are compared to the energy
dependent amplitudes which resulted from a coupled channel fit (BnGa2011-02) to
a large data set including both pion and photo-induced reactions. Significant
deviations are observed. Consistency between energy dependent and energy
independent solutions by choosing the energy independent solution which is
closest to the energy dependent solution. In a second step, the {\it known}
energy dependent solution for low (or high) partial waves is imposed and only
the high (or low) partial waves are fitted leading to smaller uncertainties.
