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In Json.Net you can use JsonPropertyAttribute with NullValueHandling=NullValueHandling.Ignore. Looks like here is no way to do this for whole class, only explicity for each class field/property.
[JsonProperty(NullValueHandling = NullValueHandling.Ignore)]
public string SometimesNull { get; set; }
FYI. There is a reasons why Json.Net by default serialize properties with null values. Take a look, may be some of this reasons applicable to your cases.