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In this paper we present radial velocity curves of AGB variables that exhibitvarious kinds of anomalies: Semiregular variables (SRVs) with typical miraperiods, SRVs exceeding the mira 2.5 mag amplitude limit, miras with secondarymaxima in their light curves, and a SRV with a long secondary period. The starswith reliable Hipparcos parallaxes from this and from previous studies areplotted in a logP-$M_{K}$-diagram. Our objects nicely follow thelogP-$M_{K}$-relations determined for the LMC. This allows the pulsation modeto be identified. While all miras fall on the fundamental mode sequence, theSRVs fall on both the first overtone and fundamental mode sequences. The SRVson the fundamental mode sequence occur at both high and low luminosities, someof them being more luminous than larger amplitude miras. This demonstratesobservationally that some parameter other than luminosity affects the stabilityof long period variables, probably mass. First overtone pulsators all showvelocity amplitudes around 4 km/s. For the fundamental mode pulsators, thevelocity amplitude shows a correlation with light amplitude. The two miras RCen and R Nor, known for their double-peaked light curves, have velocity curvesthat are quite different. The R Nor velocity curve shows no evidence of thedouble peaks, meaning that the true pulsation period is the time betweenalternate minima or maxima. There is slight evidence for a double bump in the RCen velocity curve. It is suggested that these stars are relatively massive(3-5 Msun).Comment: 13 pages, accepted for publication in A&

A study of bright Southern long period variables