On the problem of describing elements of elliptic fields with a periodic expansion into a continued fraction over quadratic fields

For all possible quadratic number fields $K$, we obtain a description of square-free polynomials $f(x)\in K[x]$ of degree 4 such that $\sqrt f$ has a periodic expansion into a continued fraction in the field of formal power series $K((x))$, while the elliptic field $\mathcal L=K(x)(\sqrt f)$ has a fundamental $S$-unit of degree $m$, $4\le m\le 12$, $m\ne11$, where the set $S$ consists of two conjugate valuations defined on the field $\mathcal{L}$ and related to the uniformizer $x$ of the field $K(x)$.