After (1) the completion of any work or purchase, (2) acquiring a sewer or water system, or both, or public parks, playgrounds, or recreational facilities, (3) contracting, as permitted by section 31-727, with other sanitary and improvement districts to acquire public parks, playgrounds, and recreational facilities for the joint use of the residents of the contracting districts, or gas or electric service lines or conduits, or (4) completion of the work on (a) a system of sidewalks, public roads, streets, highways, public waterways, docks, or wharfs and related appurtenances or (b) levees for flood protection for the district, the engineer shall file with the clerk of the district a certificate of acceptance which shall be approved by the board of trustees or the administrator by resolution. The board of trustees or administrator shall then require the engineer to make a complete statement of all the costs of any such improvements, a plat of the property in the district, and a schedule of the amount proposed to be assessed against each separate piece of property in such district. The statement, plat, and schedule shall be filed with the clerk of the district within sixty days after the date of acceptance of: The work, purchase, or acquisition of a sewer or water system, or both; the work on a system of sidewalks, public roads, streets, highways, public waterways, docks, or wharfs and related appurtenances, or dikes and levees for flood protection for the district; or as permitted by section 31-727, the acquisition of public parks, playgrounds, and recreational facilities whether acquired separately or jointly with other districts. The board of trustees or administrator shall then order the clerk to give notice that such statement, plat, and schedules are on file in his or her office and that all objections thereto or to prior proceedings on account of errors, irregularities, or inequalities not made in writing and filed with the clerk of the district within twenty days after the first publication of such notice shall be deemed to have been waived. Such notice shall be given by publication the same day each week two consecutive weeks in a newspaper of general circulation published in the county where the district was organized and by handbills posted along the line of the work. Such notice shall state the time and place where any objections, filed as provided in this section, shall be considered by the board of trustees or administrator. The cost of such improvements in the district which are within the area of the zoning jurisdiction of any municipality shall be levied as special assessments to the extent of special benefits to the property and to the extent the costs of such improvements are assessed in such municipality. The complete statement of costs and the schedule of proposed special assessments for such improvements which are within the zoning jurisdiction of such municipality against each separate piece of property in districts located within the zoning jurisdiction of such municipality shall be given to such municipality within seven days after the first publication of notice of statement, plat, and schedules. When such improvements are within the area of the zoning jurisdiction of more than one municipality, such proposed special assessments schedule and statement need be given only to the most populous municipality. Such municipality shall have the right to be heard, and it shall have the right of appeal from a final determination by the board of trustees or administrator against objections which such city has filed. Notice of the proposed special assessments for such improvements against each separate piece of property shall be given to each owner of record thereof within five days after the first publication of notice of statement, plat, and schedules and, within five days after the first publication of such notice, a copy thereof, along with statements of costs and schedules of proposed special assessments, shall be given to each person or company who, pursuant to written contract with the district, has acted as underwriter or fiscal agent for the district in connection with the sale or placement of warrants or bonds issued by the district. Each owner shall have the right to be heard, and shall have the right of appeal from the final determination made by the board of trustees or administrator. Any person or any such municipality feeling aggrieved may appeal to the district court by petition within twenty days after such a final determination. The court shall hear and determine such appeal in a summary manner as in a case in equity and without a jury and shall increase or reduce the special assessments as the same may be required to provide that the special assessments shall be to the full extent of special benefits, and to make the apportionment of benefits equitable.