In this work we solve the design of second order infinite impulse response (IIR) filters for the first time. This design is based on integer linear programming (ILP) methods. One of the underlying problem of this design is the multiple constant multiplication (MCM) problem. We propose variants to known ILP models that solve the MCM problem and illustrate our improvements with multiple examples. Our multiplierless filter design is optimal with respect to the number of adders used. Combined with MCM variants this allows addressing this optimal design in multiple contexts, in particular for a field-programmable gate array (FPGA) implementation.