Symmetry-enforced nodal cage phonons in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Th</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>BC</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math>

Exploring unique topological states in condensed-matter systems has attracted great interest especially for the topological phonons recently. Based on the unbiased structure prediction approach combined with first-principles calculations, the long-sought crystal structure of ${\mathrm{Th}}_{2}{\mathrm{BC}}_{2}$ is determined. Most importantly, we show by the symmetry analysis and the phonon tight-binding Hamiltonian that ${\mathrm{Th}}_{2}{\mathrm{BC}}_{2}$ hosts nodal surface phonons on the ${q}_{z}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}$ plane, coexisting with nodal line phonons on the ${q}_{y}=0$ and ${q}_{y}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}$ planes, consequently, forming cagelike phonons. The nodal surface phonons are protected by the screw axis ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{C}}_{2z}$, and the nodal line phonons are enforced by inversion and time-reversal symmetries, demonstrated by the codimension argument and the effective model analysis. In addition, we also investigate the phonon surface states and the isofrequency arc on the (100) surface, which benefit the confirmation of the nodal cage phonons in experiments. Our paper not only determines the long-sought crystal structure of ${\mathrm{Th}}_{2}{\mathrm{BC}}_{2}$, but also provides an ideal candidate to realize the exotic topological phonon excitations.