No private travel insurance, statutory cover is limited.
Without private travel insurance you carry a considerable risk. For visitors from the EU and the EEA the European Health Insurance Card covers the necessary treatment in the public health system on the same terms as for locals. The medically necessary transport home, however, is not covered by the statutory funds. After a ski accident these costs can be high.
Next steps: check existing insurance policies, keep the European Health Insurance Card ready and collect receipts for all costs.