Mutation-preserving generalized cluster algebras and Laurent mutation invariants

We introduce mutation-preserving generalized cluster algebras, for which the generalized cluster mutation in each direction is independent of the seed in the mutation equivalence class. We classify all the irreducible generalized cluster algebras with this property. Then, a Markov-type Diophantine equation $x^2+y^2+z^2+2yz=kxyz$ is studied, which has a structure of the mutation-preserving generalized cluster algebra. We prove that positive integer solutions exist if and only if $1\leq k\leq 5$ and determine all mutation orbits of these solutions. In particular, multiple orbits occur for each $k=1,3$, whereas the solutions form a single orbit for each $ k=2,4,5$. We also prove a conjecture proposed by Chen-Li on Laurent mutation invariants of rank $3$ cluster algebras with irreducible sign-equivalent exchange matrices, showing that every Laurent mutation invariant is essentially a polynomial in the corresponding basic invariant.