Richtmyer–Meshkov instability (RMI) plays an important role in nature and technology, including supernovae, fusion, scramjets and nano-fabrication. Canonical RMI is induced by a steady shock and impulsive acceleration. In realistic environments the acceleration is often variable. We focus on the accelerations with power-law time-dependence, identify the values of the power-law exponent for which the dynamics is Richtmyer–Meshkov type, and apply group theory to solve the long-standing problem of RMI with variable acceleration. For early-time dynamics, we find that the RMI growth-rate depends on the initial conditions and not on the acceleration parameters. For late-time dynamics, we find a continuous family of regular asymptotic solutions, including their curvature, velocity, Fourier amplitudes, and interfacial shear function, and we study stability of these solutions. For each solution, the interface dynamics is set by the interfacial shear; the non-equilibrium velocity field has effectively no motion in the bulk and intense motion near the interface. The fastest stable solution in the family has the flattened front and the quasi-invariance property suggesting the multi-scale character of nonlinear dynamics in RMI. Our results agree with available observations and elaborate new benchmarks for future studies.