Bubble growth in a Hele-Shaw cell

The general form of the solution is obtained for the problem of the development of a bubble in a Hele-Shaw channel. This solution belongs to the class of parameterized solutions characterized by the property that the partial derivatives of the mapping function are rational in the auxiliary plane. A generalized problem of the system dynamics is written for the free parameters and its complete integrability is proved. An example of the development of an asymmetric bubble with the formation of a single fiord is given.