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Burban-Drozd showed that the degenerate cusp singularities have tameCohen-Macaulay representation type, and classified all indecomposableCohen-Macaulay modules over them. One of their main example is the non-isolatedsingularity $W=xyz$. On the other hand, Abouzaid-Auroux-Efimov-Katzarkov-Orlovshowed that $W=xyz$ is mirror to a pair of pants. In this paper, we investigatehomological mirror symmetry of these indecomposable Cohen-Macaulay modules for$xyz$.  Namely, we show that closed geodesics (with a flat $\mathbb{C}$-bundle) of ahyperbolic pair of pants have a one-to-one correspondence with indecomposableCohen-Macaulay modules for $xyz$ with multiplicity one that are locally free onthe punctured spectrum. In particular, this correspondence is established firstby a geometric $A_{\infty}$-functor from the Fukaya category of the pair ofpants to the matrix factorization category of $xyz$, and next by thecorrespondence between Cohen-Macaulay modules and matrix factorizations due toEisenbud. For the latter, we compute explicit Macaulayfications of modules fromBurban-Drozd's classification and find a canonical form of the correspondingmatrix factorizations. In the sequel, we will show that indecomposable moduleswith higher multiplicity correspond to twisted complexes of closed geodesics.  We also find mirror images of rank $1$ indecomposable Cohen-Macaulay modules(of band type) over the singularity $W = x^{3} + y^{2} - xyz$ as closed loopsin the orbifold sphere $\mathbb{P}^1_{3,2,\infty}$.

Homological mirror symmetry of indecomposable Cohen-Macaulay modules for some degenerate cusp singularities